Fox was level with long-time clubhouse leader Cameron Young on nine under heading down the last but played a stunning approach shot and then sunk a 20 foot birdie.
The 39-year-old is only the second New Zealand man to win the Open, following Bob Charles in 1963 Royal Lytham.
His final round 68 left him with a total of 270, with Young - who carded a stunning 64 - on 271. American Sam Burns, the overnight leader, finished third on eight under par.
"Oh, wow. I don't really know what to think at the moment," Fox, son of former New Zealand rugby great Grant, said after being presented with the Claret Jug.
"I've got a lot of people to thank. First of all, my family, my wife and kids are in Florida at the moment; I spoke to my kids last night, and they said, bring back a trophy.
"I think this is a pretty cool one to bring back for them."
A compelling final day alongside the Irish Sea began with Burns two strokes clear on a tightly bunched leaderboard.
But as the various challengers first threatened and then fell away in a gripping drama played out over the dunes, it looked as though Young's 64 might prove decisive.
Young, who began the day in a tie for 20th, had been finished for two hours and was hitting balls on the range as a playoff looked increasingly likely.
Fox, who blasted into contention with a men's major record-equalling 62 on Saturday, seemed to be fading around the turn but found inspiration down the stretch with four birdies in his final six holes.
He could only par the par-five 17th after getting unlucky when his tee shot rolled into a fairway bunker. That meant a birdie was required on the treacherous 18th to win outright.
His next three shots will go down in New Zealand sporting folklore as he saved the best to last.
He smashed his drive down the barren fairway, then flighted a brilliant approach to 20 feet. In front of thousands of fans, he then stroked his ball straight into the middle of the cup.
World number one Scottie Scheffler finished in a tie for fourth on seven under with home favourite Tommy Fleetwood.
