Cilic was playing in his third consecutive final, having lost in Rotterdam last week after winning the title in Zagreb two weeks ago.
The 25-year-old Croat 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.
"It feels awesome, especially after waiting for a long time last night and finishing after 1 a.m.," Cilic told reporters after the three hour, eight minute final - the longest in the tournament's history.
"It wasn't easy on my nerves. Luckily I found a way."
Cilic showed no signs of fatigue as the power-hitting duo went shot for shot before the Croat claimed the first set tie-break 8-6.
Anderson, who served 18 aces to Cilic's 16, swung back in the second set where he broke the world number 29 early to take a 3-1 advantage before losing the next three games and having to rally to win the tie-break 9-7.
Cilic then fended off four break points in the deciding third set and broke Anderson's service twice to clinch the title.
"In the third (set), I just didn't put down the clamps of getting back in the match and giving myself a chance," south Africa's Anderson said.
"If I would have capitalised a little more I would have put myself in a better position to win the title."
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