The world number three, who sparked controversy by writing "Kosovo is the heart of Serbia" on a camera lens earlier this week, survived a high-octane end to the first set en route to a meeting with Spanish 29th seed Alejandro Davidovich Fokina.
"Thank you everyone for your support and presence. I hope you had fun tonight especially in the first set - me, a bit less," Djokovic said on court.
"I was not surprised, I know him very well he can play at a very high level on all surfaces. Then I played my best game. I'm very happy with that."
Under the lights of Philippe Chatrier court, Djokovic, who is bidding for a record-breaking 23rd men's singles Grand Slam title and a third Musketeers Cup, came out with all guns blazing.
He was initially too strong and too fast for world number 83 Fucsovics.
Djokovic raced to a 3-0 lead before his opponent found his stride and effectively entered the contest, setting up three break points at 4-2.
Djokovic saved them but Fucsovics battled back to level for 5-5.
The Serbian yelled at his box on his way to the bench after holding for 6-5 before breezing through the tiebreak after Fucsovics also held.
Djokovic broke his opponent's serve in the opening game of the second set and he did not look back, cruising to victory despite being broken twice more.