The victory comes roughly four weeks after Federer's last match - a title win over Grigor Dimitrov in Rotterdam, Netherlands.
"I played the best set of the whole match and that was really good".
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.
"When you do come back from injury or when you haven't played in a long time, it just takes extra effort".
The match was carried over from Saturday after rain force play to end early with Federer one set up and the scores locked at 2-2 in the second. "So naturally it's all a change".
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In the women's section at Indian Wells, Simona Halep took pride in rallying past 19-year-old Caroline Dolehide, while Petra Kvitova's remarkable win streak skidded to a halt against another U.S. teen. It was actually quite interesting. In a press conference last week before his first match, the 36-year-old addressed his short-term goals. You know, the nighttime is much more deader.
Federer, who bounced back from knee surgery to capture the 2017 Australian Open, said regaining rhythm takes time after an extended break. The surface doesn't react so much; the ball doesn't travel so fast.
While Federer was far from flawless, he did improve to an unblemished 13-0 on the year and took the next step in snaring his third title of the young season.
For the first time in six years, Federer has entered a tournament as the world's No. 1 player.