The big-serving Croat hit back from losing the opening set to record a 4-6 6-1 6-4 victory over the Spanish world number 23.
Cilic failed to convert any of his four break points in the first set and Verdasco capitalised on his only opportunity to take the set 6-4.
However, Cilic began to dominate proceedings from the back of the court in the second set and he easily levelled up the match by winning the second set 6-1.
Cilic broke Verdasco's serve midway through the deciding set, but he failed to convert two match points on the Spaniard's serve as Verdasco forced Cilic to serve out the match.
The world number 16 found himself 15-30 down, but he hit back with two unreturnable serves to set up his third match point and the win was sealed when Verdasco netted a backhand from the baseline.