The world number six produced a swashbuckling eight-under-par 63 to finish 20-under on 264, three ahead of playing partner Mickelson (65).
The 40-year-old Stenson's round equalled the lowest in any major championship and his winning total was also a record for any of the 'Big Four' tournaments.
"Wow, this will take a little while to sink in," said Stenson as he held aloft the coveted Claret Jug. "I'm still trying to find my bearings here.
"I want to thank Phil for a fantastic battle. We played some great golf and I'm delighted to come out on top."
Like two prize fighters in a ring, the two men traded blow after stunning blow, reeling off birdie after birdie to turn the last day of golf's oldest major into a two-way procession.
The rest were nowhere with American JB Holmes (69) taking third spot on 278, 11 behind Mickelson as the leading pair evoked memories of the great 'Duel in the Sun' Turnberry Open of 1977 when Tom Watson edged out Jack Nicklaus in another epic head-to-head.
Steve Stricker (69) of the U.S. was in fourth place on 279, one in front of Northern Ireland's Rory McIlroy, Tyrrell Hatton of England and Spaniard Sergio Garcia.
Mickelson showed he meant business with a first-hole birdie after drilling a majestic approach to tap-in range while overnight leader Stenson took three putts.
