Both players demonstrated some of their best tennis in the early stages with the first set remaining on serve in the opening five games, but a break of serve at 3-2 allowed Cilic to take charge of the opening set.
The Croatian then secured a second successive break to claim the set 6-2 after 38 minutes of tennis. Cilic continued that form into the second set as he broke Anderson's opening service game to take charge at 3-0.
It did appear that Cilic would claim the set in style as he broke once more to serve for the match at 5-1, but Anderson responded with a break of his own before holding serve to leave the score at 5-3.
Cilic did not need a second opportunity, however, serving out in style to book his spot in round three.