Edwards, making his 11th start on the 50-and-over tour, birdied Nos. 8, 9, 11, 12 and 13 to reach 11 under, added four pars and closed with another birdie for a 12-under 204 total on the TPC Twin Cities.
Stadler finished with a 68 for his third straight runner-up finish in the event, while Bryant closed with a 67.
The 50-year-old Edwards, who earned a career-best $262,000, won three times on the PGA Tour, the last time in 1993 in the MCI Heritage.
Edwards, playing in the final group, bogeyed the first two holes to drop five strokes out of the lead. He was still four shots back through seven holes, but charged up the leaderboard with his birdie run.
After Stadler missed a 10-foot birdie putt on 18, Edwards needed a par on the par-5 closing hole for the victory. He put his third shot within 10 feet and rolled in the birdie putt to end any drama.
Edwards had three top-fives in his first 10 events on the tour, including a second-place finish to Dana Quigley five weeks ago in the Greater Kansas City Golf Classic.
It's the sixth time the 3M has produced a first-time tour winner. Vicente Fernandez (1996), Leonard Thompson (1998), Ed Dougherty (2000), Bruce Lietzke (2001) and Wayne Levi (2003) also won their first tour titles in the event.
Curtis Strange started the round with a three-stroke lead at 10 under. Strange still led by one through nine holes before he unraveled with a triple-bogey 7 on No 10. He finished with a 76 to tie for 14th at 6 under.
Tom Kite, the 2004 winner, finished with a 64 -- the lowest final-round score in tournament history -- to tie for fourth at 9 under. Kite's bogey-free round included eight birdies. Gil Morgan (67) also finished at 9 under.