It was the biggest triumph of the journeyman's 27 years as a professional. His longtime caddie is Augusta's Buck Moore, who first started caddying as a teenager at Augusta National Golf Club and has worked numerous Masters Tournaments.
Browne's closing even-par 71 followed rounds of 64, 69 and 65 and left him at 15-under 269.
Making few mistakes, he parred the first seven holes. After a bogey dropped him into a tie with O'Meara (72), he played the last 10 holes in 1-under, while O'Meara, the 1998 Masters Tournament champion, made two costly bogeys.
Mark Calcavecchia shot 69 and was alone in third at 273. Ageless Hale Irwin had 68 was another shot back along with Joey Sindelar (70).