When play finally started, Federer set out to avenge his defeat to Delbonis in 2013 - the pair's only other ATP meeting.
Federer started quickly hammering a backhand victor to take the lone break of the first set and he would deny a pair of break chances when serving for the opener at 5-3.
That day Federer clinched the 1st of 5 US Open and the 4th of 20 Majors.
Federer is a five-time victor of the BNP Paribas Open, which ties him with Novak Djokovic for the most of all time.
The world No.1 - who has a flawless 13-0 record this year - is the defending champion and is bidding for a sixth Indian Wells title.
Federer will face 25th-seed Filip Krajinovic in the third round.
The world number one had looked well on his way to a comfortable second-round victory when rain halted play on Saturday with the second set tied at 2-2, but he did not have it all his own way a day later.
Novak Djokovic produced a puzzling performance in his loss to Japanese qualifier Taro Daniel.
Daniel defeated Djokovic 7-6 (7/3), 4-6, 6-1 in the hard-fought victory.
In a post match interview the former World Number 1 said that he felt like this was his first match on the pro circuit despite having won 12 Grand Slam titles. "Very weird", Djokovic said.
Djokovic is making a cautious return to the ATP Tour after stopping his 2017 season following Wimbledon due to a right elbow injury.
"Just struggled also a little bit with the health the last couple of weeks". And with 40-Love, you know, I just - I lost that point.
Djokovic was makings his first appearance since being stunned by Hyeon Chung at the Australian Open.
Djokovic had four aces but overall he served poorly making four double faults and winning just 65 per cent of his first-serve points.
"I was attacking when I had my chances and controlling most of the times the points", said Cilic.
"It is still early stages for Novak coming back and the first one after surgery. So talking about faster, I just think there is more bounce to it, especially if you play someone who has a lot of spin like he has and kicks the ball around a lot".
Israeli Dudi Sela beat British 21st-seeded Briton Kyle Edmund 6-4 6-4 while Luxemburg's 24th seed Giles Muller was upset by Frenchman Pierre-Hugues Herbert 6-3 7-5.