The current world number 11 started the encounter brightly and won the opening set in just under 30 minutes, courtesy of two breaks of his Argentine opponent's serve.
Zeballos provided more resistance during the second set, although it looked as though Cilic was going to seal the win when he served for the match 5-4 in front.
However, world number 39 Zeballos broke back to level up proceedings before the set went into a tie-breaker. That was also evenly contested, but Zeballos eventually forced a deciding set with an 8-6 success.
Cilic took control of the third set in the early stages when he broke at 1-0 before holding serve to move three games ahead.
Zeballos managed to respond with two holds of his own, but the Argentine was broken at 5-2 as Cilic booked his spot in the next round following just over two hours of tennis.