Fox failed to match the birdie, his putt agonizingly lipping out, and Knox put his hands to his face.
“Tough to describe how amazing this feels,” said Knox, who shot 6-under 66 in his final round.
“It’s why I play golf — all the practice days, all the misses, all the bad moments, all are taken care of with putts like that ... Making two of them from almost identical positions. I mean, that’s a bit of a bonus. Unbelievable.”
Knox is set to climb into the world’s top 50, having tied for second place at the French Open last week, and is fifth in the Race to Dubai. He has boosted his chances of making Europe’s Ryder Cup team for the match against the United States in France in September.
“To make the Ryder Cup team, you have to win tournaments,” Knox said. “This is a heck of a start but I feel I have more to go.”
Fox, who had a 68, secured a place in the British Open at Carnoustie, Scotland, later this month.
Jorge Campillo (65) was in third place, a shot further back, and defending champion Jon Rahm was in a share of fourth after a 66.
Rory McIlroy shot 71 and was 2 under overall.