Second seeds Novak Djokovic and Angelique Kerber are among those in action later on Tuesday.
In earlier matches, unseeded American Donald Young beat French 14th seed Lucas Pouille 6-4 1-6 6-3 in the men’s draw, while Japan’s fourth seed Kei Nishikori swept past Frenchman Gilles Muller 6-2 6-2.
Australian 15th seed Nick Kyrgios beat 18th seed Alexander Zverev of Germany 6-3 6-4.
In the women’s draw, American 12th seed Venus Williams beat Peng Shuai of China 3-6 6-1 6-3, and Czech third seed Karolina Pliskova led Timea Bacsinszky 5-1 when the Swiss retired.
Briton Kyle Edmund faces second seed Novak Djokovic late on Sunday.
Garcia, who was once described by Murray as a future world number one, showed impressive resilience to recover from a set down against Konta and sealed her win and a place in the last 16 with a powerful cross-court backhand.
BBC tennis correspondent Russell Fuller at Indian Wells
Konta played without her usual fluency and, although she served better in the final set, she could not take any of three break points and was outplayed emphatically by Garcia in the tie-break.
The best part of four weeks off tour resting a foot injury may explain some of the rustiness, and – like Andy Murray – Konta now has virtually two weeks of practice stretching ahead of her before she plays her first singles match in Miami.
British number two Dan Evans was beaten 6-3 6-4 by Japan’s Kei Nishikori in the second round of the BNP Paribas Open.
With temperatures at Indian Wells in excess of 105 degrees, Evans broke in the opening game but world number five Nishikori hit back immediately and captured the set in 42 minutes.
Breaks were traded again at the start of the second before Nishikori sealed the match in an hour and 24 minutes.
Briton Kyle Edmund faces second seed Novak Djokovic late on Sunday.
Following the surprise defeat of world number one Andy Murray at the hands of qualifier Vasek Pospisil the previous evening, Evans, 26, produced a spirited performance, but some pinpoint accuracy from Nishikori saw him break again in game eight of the first set.
World number 41 Evans, who beat Dustin Brown in the previous round, continued to frequently match the quality of his Japanese opponent’s groundstrokes but gradually faded as the assured Nishikori moved through to face Gilles Muller of Luxembourg in round three.
After Murray took a 4-2 lead early on, the Canadian hit back to win six successive games, claiming the first set before finally winning the second 7-5 in a tie-break, hitting a cross-court winner on his fourth match point.
“It was obviously a disappointing one as I had opportunities in the first set but I didn’t serve well enough,” Murray told BBC Sport.
“I served a few double faults, especially in the first set at important moments, which didn’t help things.
“He definitely started to play better in the second set, he was being aggressive and coming to the net and played some great reflex volleys at important moments and deserved to win.”
However, he remains in this year’s doubles alongside fellow Briton Evans as they face Dutchman Jean-Julien Rojer and Romanian Horia Tecau in round two.
Evans plays Japanese fourth seed Kei Nishikori in the singles later on Sunday. Listen to the match live on BBC Radio 5 live sports extra and online from 18:00 GMT.
Pospisil faces Dusan Lajovic in the third round of the singles after the Serbian qualifier upset 30th seed Feliciano Lopez of Spain 6-2 4-6 7-6.
Elsewhere, French seventh seed Jo-Wilfried Tsonga was beaten by Italy’s Fabio Fognini but there were wins for third seed Stan Wawrinka, 10th seed Gael Monfils and 11th seed David Goffin.
BBC tennis correspondent Russell Fuller in Indian Wells
World number one or not, Murray has often struggled in the desert. His serve let him down – he hit seven double faults and was broken four times in a row – and was ultimately second best to a man who is having a great year against the Brits.
Pospisil may be a qualifier ranked 129 in the world but his serve-and-volley game is mightily effective, as Dan Evans and Kyle Edmund learned to their cost in last month’s Davis Cup tie with Canada.
Unusually for Murray, he is now out of the singles but still in the doubles so he will stay in Indian Wells to partner Evans and to spend “lots of time” on the practice courts.
Great Britain will travel to the Black Sea city of Constanta for their Fed Cup World Group II play-off against Romania.
The home city of Romanian number one Simona Halep will host the tie on outdoor clay on 22-23 April.
Britain are looking to return to the elite level of the competition for the first time since 1993, but will go into the tie as heavy underdogs.
Halep, the world number four, has already said she will play in the tie.
Romania have four other players in the top 100.
GB captain Anne Keothavong’s team kept their promotion hopes alive last month with a 2-1 win over Croatia. and while she will hope to call on Johanna Konta, the world number 11 is not at her best on clay.
British number two Heather Watson is currently ranked 108 but has a strong Fed Cup record with 25 wins and only seven losses.
It is the third time Britain have reached the World Group II play-offs in the past six years, with the team then captained by Judy Murray losing to Sweden and Argentina in 2012 and 2013 respectively.
The 29-year-old beat Fernando Verdasco to win the title for the first time.
But a fourth-round defeat by Mischa Zverev at the Australian Open in January means Murray has ground to make up on Roger Federer, Rafael Nadal, Grigor Dimitrov, Jo-Wilfried Tsonga, Dominic Thiem and David Goffin in the 2017 rankings.
He is, however, likely to retain the number one ranking until at least the French Open.
“I felt like I wasn’t a bad player just because I lost a match at the Australian Open,” he told BBC Sport after a 16-hour flight from Dubai to Los Angeles.
“Australia wasn’t my tournament but I took a break after that, chatted to my team about things that I needed to work on, worked on them, and got to Dubai early.
“I played some good stuff where I hadn’t played well in the past. So that gave me a bit of a boost coming here, which is also a place where I haven’t played my best.”
Ready for the challenge
Murray was the runner-up to Rafael Nadal at Indian Wells in 2009, but in the past six years has suffered early defeats at the hands of Donald Young, Guillermo Garcia-Lopez and – in last year’s third round – Federico Delbonis.
The thin desert air makes the ball fly and jump off the court, and last year played havoc with the Scot’s serve. He held back, for fear of missing, and was beaten in the second match he played.
His preparation had also been far from ideal. On the Sunday before an event due to be staged outdoors in 30 degree desert heat, Murray spent four hours and 54 minutes to beating Kei Nishikori in the Davis Cup on an indoor court eight time zones away in Glasgow.
However, this year’s warm up in Dubai was much preferable and Murray was enthusiastic when talking about Great Britain’s Davis Cup quarter-final in France.
That tie was secured in February when Britain beat Canada 3-2 without their leading player and may be held after the Miami Masters, which follows Indian Wells.
“It will have been a long stretch, but to get matches on clay is a positive thing – and my team are more pro it,” Murray said.
“If I’d gone to Canada, it would have been bad news because physically I was not ready. I was struggling a little bit with the illness so it was a good thing I didn’t go.”
Having overcome shingles, Murray now has the awkward desert conditions to overcome, but has been given a favourable draw in Indian Wells.
While his quarter is far from treacherous, the bottom quarter includes Novak Djokovic, Nadal, Federer, Juan Martin del Potro, Nick Kyrgios and Zverev.
“If you look at our leadership team at the moment within the LTA [nine people on the executive team plus head coaches Leon Smith and Jeremy Bates] – there is only one woman, and that’s the lady who runs the Human Resources department.
“So in that very important team – the decision-making team – there is no female tennis voice and to me that is completely wrong: something that really needs to be addressed. We need a much better balance.”
Inspiring more girls to continue playing the sport as they approach their teenage years is not a problem unique to tennis, and Murray believes female coaches are a critical part of the equation.
Murray designed the Miss-Hits programme, which is aimed at girls aged between five and eight, and at the beginning of February launched a female coaching initiative called She Rallies. Both programmes are run in partnership with the LTA.
“I tried hard when I was Fed Cup captain to grow the women’s side of the game,” she said.
“We don’t have anywhere near enough women coaches and I do believe there is a correlation between the number of female coaches and our ability to retain girls in competitive sport.
“Women so much better understand how girls think and behave and what their needs are. Teenage girls, in my experience, are not going to open up about their fears – such as issues with their bodies – to male coaches.”
The disparity between the number of male and female coaches is starkly illustrated in the professional game. Britain’s top three women – Johanna Konta, Heather Watson and Naomi Broady – have taken on male coaches in recent months – principally because there is a much bigger pool to choose from.
“My experience of working with women coaches is that there aren’t egos,” Murray continued.
“They are much more willing to work together, and network and share. And I think if we can use that as a starting point, then we get more women doing things together and we can really start to make some inroads.”
The LTA says it has insisted on there being at least one woman on the shortlist for the last three executive vacancies, and is proud of the fact that the nine women who have returned from maternity leave in the past 18 months have all been offered flexible working. A specific strategy aimed at women and girls is set to be launched later in the year.
And as for increasing the number of women in the senior leadership team, chief executive Michael Downey says it is a priority for the LTA, but that it will not happen overnight.
“Given that our sport is pretty gender-balanced, we want to have more gender balance in leadership roles,” Downey told the BBC.
“Change like that can take some time to get there, but we’ve got to keep working on it: it’s the right thing to do.
“We spend a lot of time on the key hires, and hopefully more often or not there will be qualified women who give us an opportunity to achieve some of those metrics moving forward.”