Djokovic looked rusty in his 7-6 (3), 4-6, 6-1 defeat against Daniels at the BNP Paribas Open in Indian Wells.
Federer, who earned his fifth title at Indian Wells a year ago, saved a set point in the second-set tiebreaker and advanced to face Filip Krajinovic on Monday.
On his return from right elbow surgery, Djokovic looked a pale shadow of his former self as he went down 6-7 (3-7), 6-4, 6-1 to make a quick exit from a tournament he has won five times in the past.
The 30-year-old took six months off after Wimbledon last year and then lost to Hyeon Chung in the Australian Open fourth round in January before going under the knife in Switzerland shortly after.
The 10th-seeded Djokovic acknowledged his poor performance against the world's 109th-ranked player.
Other seeded men advancing were: No. 7 Kevin Anderson defeated Evgeny Donskoy 7-5, 6-4, No. 12 Tomas Berdych rolled past German Maximilian Marterer 6-1, 6-4 in just over an hour.
"But, you know, after an opponent like this, to come back from two breaks down, actually, for me it was nearly impossible". "Nerves were there. I made so many unforced errors".
Verdasco's first serve proved almost unplayable in the deciding set of the second round match as he improved his career head-to-head record against Dimitrov to 2-2.
"Very weird", Djokovic said.
Daniel's third-round opponent is Argentina's Leonardo Mayer, who had been scheduled to play Japan's Kei Nishikori in the second round before the world No. 25 withdrew through illness.
"I just completely lost rhythm, everything, struggled a little bit with the health the last couple of weeks". It was the former world No.1's first match since returning from an injury, which forced him to undergo a surgery earlier in February.
"I have been sick for over a week and tried everything I could to get better".
The Swiss superstar is the defending champion at Indian Wells.
Daniel, who is ranked 109th in the world, advanced to the third round where he will face Gael Monfils of France who outlasted American John Isner 6-7 (5/7), 7-6 (7/3), 7-5.
Delbonis actually had a set point during the tiebreaker in the second set, but Federer thwarted off the threat and went on to win.