The Argentine broke midway through the first set to draw first blood against his big-serving opponent, who is ranked number 22 in the world.
Despite a niggling knee problem, Del Potro served out to claim the opener and then broke again early in the second to take firm control of the contest at Roland Garros.
Cilic responded with a break of his own before the ninth seed wasted a fantastic opportunity to take a two-set lead when he squandered seven match points in the 12th game.
It wasn't to prove costly though as Del Potro won the resultant tie-break 9-7.
Seemingly buoyed by his narrow success, the former US Open champion stormed the third set, dropping only one game as he secured a 6-3 7-6 6-1 triumph.