The 25-year-old Croatian was pushed to the limit in a topsy-turvy match of powerful serves that lasted a little over three hours.
After they split the first two sets in tie-breaks, Cilic fended off four break points in the deciding third set and broke Anderson's service twice to clinch the title.
South Africa's Anderson, who was looking for his second title in Delray Beach in three years, blasted 18 aces to Cilic's 16.
Cilic had entered the final having had little time to recover from his semi-final with American John Isner that was not completed until early on Sunday after being delayed for three hours due to rain.
He showed no signs of fatigue as the power-hitting duo went shot for shot before Cilic claimed the first set tie-break 8-6.
Anderson swung back in the second set where he broke Cilic early to take a 3-1 advantage before losing the next three games and having to rally to win the tie-break 9-7.
(Writing by Jahmal Corner in Los Angeles; Editing by Greg Stutchbury)
