Cilic, who pounded out 19 aces, had his Czech opponent under pressure right from the start, breaking Berdych at the first opportunity in the first two sets to grab a 2-0 lead.
Berdych turned the tables on the 14th-seeded Croatian in the third and raced out to a 3-0 lead but Cilic, relying on his booming serve, broke back and forced a tiebreak before clinching the match with a thundering ace and forehand winner on the final two points.
Up next for Cilic is the winner of the clash between five-time US Open champion Roger Federer and Frenchman Gael Monfils, who play later today.