The Serbian eventually secured a 6-3 6-2 7-6(2) 6-4 victory over Cilic to ensure his place in the fourth round alongside Roger Federer.
Djokovic suffered a wobble in the opening set when he lost his second service game, but he was able to bounce back by grabbing a double break and taking the lead on the clay court.
Cilic was able to hold off the Serbian's attack in his first service game in the second set, but Djokovic eventually got his way by taking a break and 2-1 lead.
Djokovic was made to work for his set point after Cilic brought him back to deuce three times before the former number one ranking took a two-set lead.
A break apiece resulted in the third set moving into a tie-break, which Cilic was able to win comfortably 7-2 to keep himself in contention.
Djokovic took control of the fourth set by taking a 4-1 lead, but Cilic bounced back with a break and eventually levelled proceedings on 4-4 before Djokovic dealt the final blow with a break to win the match.