Five strokes behind the pacesetting Rollins overnight, Watney birdied three of the last six holes on the picturesque South Course to seal victory with a four-under-par 68.
After sinking a 40-foot birdie putt from the fringe of the 16th green to move into a tie for the lead, Watney also birdied the par-five 18th for an 11-under total of 277.
Rollins, three strokes in front with five holes to play, had to settle for second place after closing with a topsy-turvy 74.
Colombian Camilo Villegas, who narrowly missed a 40-foot eagle putt at the last, closed with a 72 to share third place at nine under with American Lucas Glover (68).
Three ahead overnight, Rollins made a stumbling start with consecutive bogeys, three-putting at the first and missing the green to the right with his approach at the par-four second.
That trimmed his lead to two, playing partner Villegas having also bogeyed the second after finding a poor lie in the left rough off the tee and failing to reach the green in two.
Rollins made his first birdie of the day at the par-five sixth, chipping up from just short of the green to three feet and knocking in the putt to regain his three-shot cushion at the turn.
Rollins Stumble
Glover closed to within a stroke of the lead with birdies at the 10th and 13th before Rollins again stumbled.
After finding a fairway bunker off the tee at the 12th, Rollins found the right rough with his approach before chipping to nine feet and narrowly missing the par putt.
With his lead shrunk to one, Rollins immediately responded in potential champion style with a brilliant eagle at the par-five 13th.
His second shot there ended up 21 feet from the hole and he coolly rolled in the right-to-left breaking putt before pumping his right fist in celebration.
However, Rollins bogeyed two of the next three holes to relinquish the outright lead.
On 14, he again missed the fairway off the tee and found tangly rough just off the green with his approach before ending up in a plugged lie in a greenside bunker at the par-three 16th.
Watney, who had trailed Rollins by two standing on the 16th tee, rammed in his long-range birdie putt to move into a tie for the lead at 10 under.
Both players parred the 17th before Watney reached the green in two at the last and coaxed his approach putt to within three feet of the cup.
Rollins, whose second shot there ended up in a greenside bunker, splashed out to 12 feet but missed his birdie attempt before Watney calmly rolled in his putt to secure the title.
