Key burdie putts of 40 feet and 25 feet helped Fowler overcome two tee shots into the water.
He started the day with a four-shot lead and with five birdies, four bogeys and a double-bogey finished with a 12-under total of 268 -- four strokes ahead of Morgan Hoffmann and Gary Woodland.
"My putter saved me," said Fowler, who was in the water off the tee en route to a double bogey at the sixth, and shockingly in the water again off the tee at 17, where he salvaged a bogey.
He dropped another shot at the par-five 18th, where his third shot found a greenside bunker.
Despite the ugly ending, the round saw Fowler win for the first time when holding the 54-hole lead on the PGA Tour, after four prior misses.
His lead had dwindled before he rolled in a 40-footer at the 12th and a 25-footer at the next, and his advantage was up to five strokes when he birdied 16.
Woodland had applied some early pressure, but he finished with back-to-back bogeys to settle for a 69 and a share of second on 272 alongside Hoffmann, who carded a 68.
England's Tyrrell Hatton, who started the day in second place, carded a two-over 72 to finish in a large group sharing fourth on 273.
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