Federer returned on Sunday afternoon and needed an hour to close out his second-round victory 6-3 7-6 (8/6).
However, 25-year-old Daniel dominated the third and final set to take it 6-1.
British number one Kyle Edmund was beaten in straight sets by Israel's Dudi Sela 6-4 6-4 in his first match since reaching the Australian Open semi-finals.
"I think the beginning of the matches are always important, get off to a good start", Federer said.
With rivals Rafa Nadal, Stan Wawrinka and Andy Murray electing not to enter the tournament, and Novak Djokovic and Kei Nishikori both out of the running, the path appears clear for the Swiss to win a record sixth Indian Wells title.
Mark Hughes 'favourite for Southampton job'
Southampton have won only once in their last 17 EPL outings, though the team have qualified for the quarter-finals of the FA Cup . Southampton fired manager Mauricio Pellegrino on Monday in a bid to stop the team sliding out of the English Premier League.
But Djokovic fell at the first hurdle with a shock loss to Japanese qualifier Taro Daniel, who sent the out-of-sorts Serb tumbling with a hard-fought 7-6 (7/3) 4-6 6-1 victory. "I mean, I just completely lost rhythm, everything".
Another surprise victor was Cyprus' Marcos Baghdatis, who beat Argentina's Diego Schwartzman, who is seeded 14th, 7-5, 6-4.
Federer, who had looked slightly off his game in his rain-disrupted third round match against Federico Delbonis, was his dominant self on Monday, winning 89 percent of his first service points while cracking six aces to just one double fault. "And even though you don't have pain, you are still thinking about it, because it has been something that I have been feeling and dragging for over two years", Djokovic said. "She's the best player I have ever played, and it was the biggest court I have ever played on".
Former number one and tenth-seed Angelique Kerber rallied past Ekaterina Makarova 3-6, 6-4, 6-2 and Petra Martic upset last year's French Open champion Jelena Ostapenko 6-3, 6-3.