It was the highest closing score by a winner since Vijay Singh finished with a 4-over 76 in the 2004 PGA Championship, and the highest in a non-major since Fred Couples had a 5-over 77 in the 1983 Kemper Open.
Bowditch finished at 8-under 280 at TPC San Antonio and earned $1,116,000.
"Every time I got out of check, looking ahead to the Masters and winning golf events and making my speeches before I was finished, I had to pull myself in check every time," said Bowditch, wearing a green shirt. "And it happened a lot today."
Bowditch, based in Dallas, entered the week 339th in the world and had only two top-10 finishes in eight years on the tour. He won once on the Australasian circuit and twice on the Web.Com Tour.
"He's been a battler. He's gone through a lot in his life," said John Senden, a fellow Australian who won the Valspar Championship two weeks ago.
Will MacKenzie and former BYU golfer Daniel Summerhays tied for second. MacKenzie shot 70, and Summerhays had a 71.
In Carlsbad, Calif., Anna Nordqvist rallied to win the Kia Classic at Aviara for her second victory in her last four LPGA Tour starts.
Two strokes behind leaders Cristie Kerr and Lizette Salas entering the round, Nordqvist closed with her second straight 5-under 67 for a one-stroke victory over Salas.
Nordqvist won the LPGA Thailand last month to end a five-year victory drought, holding off top-ranked Inbee Park.