The Wimbledon champion had been the second seed at the tournament, but got off to a slow start, finding himself 4-1 down in the first set following a poor start on serve from the Briton.
With a comfortable lead, Cilic was able to serve out the first set without facing a single break point.
Murray lost an early break again in the second as he struggled for find his rhythm. He created several chances to get the set back on serve, but his opponent was able to save all four break points against him.
At 5-4, Cilic brought up two match points on his own serve, and was able to wrap up the 6-3 6-4 win at the first time of asking with a forehand winner.