Choi and David Toms went into a play-off after McDowell - three clear with one to play in the third round earlier in the day - double-bogeyed the 18th and then shot a closing 79.
Choi trailed Toms by one with three to play, but the American - without a win for more than six years and 75th in the world - bogeyed the long 16th after hitting his second into the lake.
Choi then made a ten footer for birdie at the fearsome par-three 17th, but Toms made a brilliant 17-foot birdie putt at the last to tie Choi on 13 under par.
McDowell, trying to add to his US Open Championship win last summer, ended up in 33rd place.
Luke Donald was the top European, but came up short in his bid for the victory that would have taken him to World Number One for the first time.
He was still in with a chance at three behind after a two-putt birdie on the 16th, but missed an 11 foot chance at the next and, needing a birdie at the last to retain an interest, did well to make par after a wild drive to the right.
Paul Goydos, who lost a play-off to Sergio Garcia in the event three years ago, came third on 11 under and Donald was one further back with Nick Watney.
That makes it seven top-ten finishes in a row for WGC-Accenture Match Play champion Donald - and 12 in his last 13 starts going back to last September.
The day started with the completion of the third round and when McDowell birdied the 16th and 17th he was six under for the round and firmly in command.
But he had a bad break on the last when his approach kicked off the side of a mound right across the green and into the lake.
After signing for a 68 he was leading by one and still in high spirits, but that soon changed.
It all started to go wrong on the 393 yard sixth, moments after he had the bonus of a 50 foot birdie putt on the previous green.
McDowell carved his tee shot into the trees, then, after bogeying that hole, hooked his drive down the seventh into the lake and dropped another shot.
Worse was to come. Going for the green in two on the long ninth he hooked again into a small bush and, seeing no better option than to have a thrash at the ball, moved it only a few feet.
The next shot found rough and although he got up and down, it meant a bogey six.
Not that he was out of it at that point - he turned three behind Toms - but after finding the water at the short 13th and bogeying yet again his hopes were hanging by a thread.
Toms had an 18 footer for victory at the first extra hole - the short 17th - but incredibly three-putted and it was Choi who triumphed with two putts from 40 feet.