Cilic, the 2014 US Open champion, blasted 47 winners including 13 aces during the match, which lasted two hours and 13 minutes.
The Croatian world No. 4 is set to play Poland's Hubert Hurkacz, who beat the US' Tennys Sandgren 6-2, 6-2, 3-6, 6-3.
Meanwhile, Germany's Jan-Lennard Struff managed to overcome Evgeny Donskoy of Russia 6-1, 6-3, 6-0 in just one hour and 20 minutes.
Struff's opponent in the second round is to be the winner of the match between US player Steve Johnson and France's Adrian Mannarino.