Neither player was able to break their opponent's serve in the slugfest, which lasted almost three hours, with each set decided by tiebreakers.
"I've been on the wrong end of a final that close before - maybe not three tie-breakers close, but super close, having match point, having a lot of chances - so it feels good to be on the other end of that," said Isner.
"I never wavered. I believed that even at 0-40 that I could still hold on and was able to come up with the goods."
Anderson, a 6-foot-8 (2.03m) right hander from Johannesburg chasing a third career title, won the first tiebreaker and had 11 chances to break Isner's serve but failed to convert any of them.
He hit 21 aces and only faced one service break on his own serve and had two match points in the third set but was unable to seize either chance.
"I was holding easily and trying to give myself chances, which I did. Just wasn't able to capitalise on them. That was the difference today," said Anderson.
"He stays in there. He serves well, he plays well when it matters, and he definitely played two better tie-breaks than me."
(Reporting by Julian Linden in New York; Editing by Gene Cherry/Peter Rutherford)
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