Federer, who earned his fifth title at Indian Wells previous year, saved a set point in the second-set tiebreaker and advanced to face Filip Krajinovic on Monday. After a two-hour delay it was decided they would resume on Sunday.
"I never sit down with Rafa and say "let's put it on the line and talk about who is the best of all time", Federer said.
A backhand victor gave Federer the only break of the opening set but, mid-way through the second, rain forced them to depart. Upon resumption on Sunday, Delbonis did well to force a tie-break, but Federer was too strong in the end.
Federer won the final three points to take the match by forcing Delbonis to dump all three of those shots into the net.
World number one and defending champion Roger Federer, meanwhile, was unable to complete his opening match due to rain.
World number four Grigor Dimitrov slumped to a shock 7-6 (7/4), 4-6, 6-3 defeat to Spain's Fernando Verdasco in his opening match of the ATP Indian Wells Masters on Saturday.
It was another snag in the return for the world number 13 Djokovic who admitted he lacked composure while playing in just his second event of 2018 and first since having surgery. "Very weird", Djokovic said.
Notre Dame basketball misses out on NCAA Tournament
The Pirates earned the NIT bid after earning the top-seed in the MEAC Tournament with its share of the regular-season title. Sophomore guard Tony Carr was selected first-team All-Big Ten and USBWA All-District and to the Big Ten All-Tournament Team.
"I just completely lost rhythm, everything, struggled a little bit with the health the last couple of weeks". As I am the defending victor, there is also a little bit of pressure.
On his highly-anticipated return to the tour after right elbow surgery, Djokovic looked a shadow of his former self. He lost to Chung Hyeon in the Australian Open fourth round in January before undergoing the surgery.
The 12-time major victor led 5-2 in the opener, but Daniel fought back well to clinch the first set on a tie-break, and although Djokovic pulled level, 61 unforced errors from the five-time champion saw Daniel clinch victory after two-and-a-half hours.
Djokovic had four aces but overall he served poorly making four double faults and winning just 65 percent of his first-serve points.
"It feels insane. I mean, I'm still in shock", Anisimova said.
"When you do come back from injury or when you haven't played in a long time, it just takes extra effort".
Gael Monfils upset No. 15 seed and 2012 runner-up John Isner 6-7 (5), 7-6 (3), 7-5.