In their only other ATP meeting in 2013, Delbonis won on clay in Hamburg 7-6 (7), 7-6 (4).
Rain barred Federer from completing the first match of his title defence at the season's opening ATP Masters 1000 tournament on Saturday.
Djokovic was makings his first appearance since being stunned by Hyeon Chung at the Australian Open.
With his triumph on Monday, Federer, victor of the 2018 Australian Open (20th Grand Slam title) and the Rotterdam tournament, extends his winning streak this year to 14-0.
Roger Federer, of Switzerland, returns a shot against Jack Sock during an exhibition tennis match in San Jose, California.
The world No 100, however, responded well and took the initiative with an early break before a Mannarino double fault handed Chardy, who plays at Patrick Mouratoglou's tennis academy, a second break of the final set before he served out for the win. 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 yesterday.
Djokovic scrambled to win the second set, but looked rusty on stadium court, making 62 unforced errors compared to 29 for Daniel.More news: Aston Villa boss says only an 'almighty collapse' will deny Wolves
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"Very weird. I completely lost rhythm", admitted the Serb.
Israeli Dudi Sela beat British 21st seed Edmund 6-4 6-4, while Luxemburg's 24th seed Giles Muller was upset by Frenchman Pierre-Hugues Herbert 6-3 7-5.
On the women's side, top-ranked Simona Halep came back to top 19-year-old Caroline Dolehide 1-6, 7-6 (3), 6-2 in just over two hours. So it was definitely nerve-wracking kind of, but I was enjoying it so much out there and I was playing my best.
Federer is back on court for a third straight day against Djokovic countryman Filip Krajinovic.
Eighth seed Jack Sock, 18th seed Sam Querrey, 32nd seed Milos Raonic all progressed into the third round.
Yuki Bhambri of India pulled off a surprise when he toppled ninth-seeded French ninth seed Lucas Pouille 6-4 6-4.
The win sets up a third-round showdown with Spaniard David Ferrer, who defeated Tennys Sandgren of the United States 6-2, 7-6 (7-3).