The former world number three, unseeded this year at Roland Garros after falling to 46th in the rankings, produced a gutsy display laced with pure shot-making to win 6-7 6-4 4-6 7-6 6-3.
Dimitrov moved 5-2 ahead in the decider with two breaks, but Cilic was not going out without a fight and saved a match point with a booming forehand as he repaired some of the damage.
Cilic netted a backhand in the next game, however, to hand Dimitrov a notable victory.
Dimitrov has never been past the third round at Roland Garros and if he is to extend his stay beyond that this year, he will have to do it the hard way.
His next opponent is 2015 champion Stan Wawrinka.
(Reporting by Martyn Herman; Editing by Andrew Cawthorne)
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