After another 90-minute delay, the organisers suspended play for the night.
The defeat was just the fifth time in the past decade former world number one Djokovic has lost against a player outside the top 100 players in the world.
The world No.1 - who has a ideal 13-0 record this year - is the defending champion and is bidding for a sixth Indian Wells title. The 24-year-old is leading the ATP World Tour with the most wins at 17-3.
Spanish veteran Fernando Verdasco used his powerful serve to upset third seed Grigor Dimitrov 7-6 (4) 4-6 6-3 at the BNP Paribas Open on Saturday while Roger Federer's tournament debut was hampered by rain. He won the final three points to take the match by forcing Delbonis to dump all three of those shots into the net.
On Monday, the third round wraps up with Federer back on court against Paris Indoor finalist Filip Krajinovic.
Federer and Djokovic both came into the tournament seeking to become the first six-time victor of the event.
Daniel, who is ranked 109th in the world, advanced to the third round where he will face Gael Monfils of France who outlasted American John Isner 6-7 (5-7) 7-6 (7-3) 7-5.
Roger Federer has stressed that he is not surprised after Novak Djokovic crashed out in the early stages of the 2018 Indian Wells and the world No.1 has backed the Serbian star to "only get better from here". The Spaniard once ranked No. 7 in the world is now ranked No. 39 said: "Well, I think that the match kind of, like, went up-and-downs a little bit, at least the two first sets with a break that he made me at the beginning".
"It feels insane. I mean, I'm still in shock, " Anisimova said. "Very weird. I mean, I just completely lost rhythm, everything". "Just struggled also a little bit with the health the last couple of weeks".
Djokovic bowed out to South Korean Chung Hyeon in the fourth round of the 2018 Australian Open.
"To me, it's not that surprising". Honestly making - I think, if I'm not wrong, in two games two double faults on each one, and giving him the possibility to go 3-0 in the score with two breaks.