The 35-year-old produced a faultless final round, hitting a two-under 69 with 16 pars and two birdies at Carnoustie.
Xander Schauffele was the only man within reach of Molinari heading down the 18th green, but having bogeyed 17, the American needed an eagle to tie, and could only manage a par.
The previous best result by an Italian at one of golf's four Major tournaments came courtesy of Costantino Rocca, who finished second at The Open in 1995.
THE OPEN FINAL LEADERBOARD
1. Francesco Molinari (-8)
2. Kevin Kisner (-6)
3. Rory McIlroy (-6)
4. Justin Rose (-6)
5. Xander Schauffele (-6)