Federer closed out the set on serve and pushed Delbonis to the brink of elimination in the process; however, he had to come back from a 15-40 deficit in order to do so.
Federer staved off a set point in the tiebreak to beat Delbonis 6-3 7-6 (6).
Djokovic, who gained a first-round bye, ought to have won the opening set but squandered a break point at 5-2 and the chance to clinch it on serve a game later. After another 90-minute delay, the organisers suspended play for the night.
Federer is making his 17th appearance in the California desert and is assured of remaining world number one if he makes it to the semi-finals in Indian Wells.
Sixth seed Juan Martin Del Potro took just under 70 minutes to beat Australian Alex de Minaur 6-2 6-1 and Australian Open runner-up Marin Cilic is also through after a 7-5 6-3 straight sets win over Marton Fucsovics.
Federer staved off a set point in the second set tie-break to beat Delbonis 6-3 7-6 (8-6) and he will face 25th-seed Filip Krajinovic in the third round on Monday.
Novak Djokovic produced a puzzling performance as the former world No 1 lost to Japanese qualifier Taro Daniel in the second round at the Indian Wells Masters on Sunday. He returned for the Australian Open with a revamped, abbreviated service motion created to relieve pressure on the elbow and showed flashes of fine form before eventually losing in straight sets to Hyeon Chung in the fourth round. The 30-year-old committed 61 unforced errors in the two-and-a-half hour match.
He has only dropped 3 of 34 sets this year.
Alan Pardew to hold West Brom crisis talks after seventh successive defeat
I don't care whatever job or workplace you're in, if you're under pressure and you don't want to fight, don't bother turning up". The club have won only three times in 38 matches, and the empty seats at the Hawthorns told of apathy.
"I just completely lost rhythm, everything", Djokovic told reporters afterwards.
The win sets up a third-round showdown with Spaniard David Ferrer, who defeated Tennys Sandgren of the U.S. 6-2, 7-6 (7-3).
Roger Federer wins yet again this week.
"We wondered post surgery what it was going to look like, how his elbow was going to react to the strain and stress, but it was his heart and head that didn't react the right way in this match". "I'd like to congratulate my opponent for fighting and showing great, great spirit on the court", said Djokovic. Daniel went toe-to-toe with Djokovic as the match wore on, matching the Serbian's aggression. Djokovic even admitted that he lacked rhythm in the game.
"It feels insane. I mean, I'm still in shock", Anisimova said.
"Definitely, I think it's motivating", Anisimova said in an interview last week. I was - I felt close, but, you know, didn't been able to come back.
Daniel, a 25-year-old ranked 109th in the world, hadn't won a main-draw ATP World Tour event this year before Sunday.
Seeded losers included Italian No. 16 Fabio Fognini, 4-6 7-6 (2) 6-4 against Jeremy Chardy of France, and Spaniard Albert Ramos-Vinolas, hammered 6-0 6-3 by young Croat Borna Coric.