Bowditch, 30, produced a four-over 76 in his final round at the TPC San Antonio, but that was enough in tough conditions, as none of his nearest challengers threatened his eight-under 280.
Will MacKenzie could only manage an even-par 70 to finish second and Daniel Summerhays (71) was alongside him a shot back at seven under.
Fellow Americans Matt Kuchar and Andrew Loupe went three-over 75 for the day and were tied for fourth at six under.
It opened it up for Bowditch, who held his nerve late in the round and secured the victory by bogeying the par-five 18th with a six-inch tap-in.
The win was Bowditch's first on the PGA Tour, with the Australian having overcome depression and alcoholism earlier in his career.
Victory guaranteed him a place at the Masters at Augusta starting April 10 and earned him $1.116million.
Bowditch started his round with a bogey at the second and a double bogey at the fourth and just when he appeared to be steadying with a birdie at five, another bogey followed at six.
A string of six consecutive pars set him up and, as none of his nearest rivals threatened, that was enough to emerge victorious, despite two bogeys and a birdie on his final six holes.
MacKenzie holed three birdies and a bogey in his round and Summerhays mixed four birdies with three bogeys.
Kuchar&aposs three bogeys on five holes to start the back nine cost him, while a double bogey at the seventh hurt Loupe.
Americans Brendon Todd (68), Jerry Kelly (71) and Jim Furyk (71) were tied for sixth at five under, a shot ahead of Jordan Spieth (71).
Australian Geoff Ogilvy (73) was alongside Americans Charley Hoffman (70), Pat Perez (77) and Kevin Na (76) in a share of 11th.