While Nole was playing his first tournament since his Australian Open win in January, he also went into the match with an 8-1 head-to-head lead.
Monfils has a 13-2 career record against Kohlschreiber.
"Things to work on, without a doubt", Djokovic said.
Her win over Halep, a former world No. 1 and victor at Indian Wells in 2015, comes after she dispatched 2018 finalist Daria Kasatkina in the second round and former French Open champion Jelena Ostapenko in the third.
"We didn't see the real Novak Djokovic here".
It was a tough day for seeded players at the BNP Paribas Open on Tuesday, with Novak Djokovic and Naomi Osaka, the world's top-ranked players, ushered out in the desert.More news: Dolphins Trade Ryan Tannehill To Titans
"She was just playing so well", said Osaka.
Nadal, a three-time champion at Indian Wells, did not face a break point in the match.
In a match that resumed after the rain stopped play just one game in on Monday night, Kohlschreiber said the warmer daytime conditions suited him better, and Djokovic agreed.
"Take care and good night, know this the good life, hot tubs and court time, Thursday we back alright!" he rapped at the end of the song, which got a roar of approval from the audience.
After Djokovic held at love, Kohlschreiber held his nerve to finish it off, gaining a match point when Djokovic's lunging backhand went wide before Kohlschreiber fired a sideline-skimming forehand. "Honestly, at a time like this with that scoreline, I would usually feel very depressed and sad".
World number four Roger Federer needed under an hour to defeat fellow Swiss Stan Wawrinka 6-3 6-4 and will next face Briton Kyle Edmund, who defeated Radu Albot in straight sets earlier in the day. The 23rd-ranked Bencic served and returned better.More news: UEFA to investigate Neymar's VAR criticism after PSG defeat
"Not easy mentally to refresh your mindset and concentration and get the right intensity on the court so quickly after disappointing loss in singles", Djokovic said.
Nadal, a three-time victor of the Indian Wells title, next faces Serbian qualifier Filip Krajinovic, who ousted 14th-seeded Russian Daniil Medvedev 6-3, 6-2.
In addition to the top seed falling on the women's side of the draw, the second seed, Simona Halep lost to teenager Marketa Vondrousova 6-2, 3-6, 6-2.
Philipp Kohlschreiber in action at Indian Wells on 12 March 2019.
Poland's 67th-ranked Hubert Herkacz upset sixth-seeded Kei Nishikori of Japan 4-6, 6-4, 6-3 to reach the last 16 of a Masters 1000 event for the first time.
Bencic, who beat four top-10 players on the way to winning a title in Dubai last month, now plays world number five Karolína Plíšková of the Czech Republic, who beat Anett Kontaveit of Estonia 7-6, 4-6, 6-2, in the quarter-finals.More news: Valve releases Steam Link Anywhere for PC