Knox, whose only previous European Tour win came at the WGC-HSBC Champions in 2015, drained a putt from around 40 feet on the 72nd hole to complete a round of 66 and join Fox at 14 under.
After Fox had failed to convert a short birdie chance on the last hole of regulation play, Knox then remarkably found the target from long range once again when the pair returned to the 18th in a play-off.
Fox was again much closer in two following a huge drive, but his putt for a three lipped out, meaning Knox could celebrate a win that significantly boosts his hopes of making Europe’s Ryder Cup team.
The Scot finished tied second last week in the French Open at Le Golf National - the course that will host the Ryder Cup in September.
“Obviously I needed a stretch like this,” said Knox regarding his hopes of making Thomas Bjorn’s team.
“To make the Ryder Cup team you have to win tournaments. This is a heck of a start, but I feel like I’ve got a lot more to go.
“It’s tough to describe how amazing this feels.”
Knox also qualifies for The Open as a result of his triumph, along with Zander Lombard and Andy Sullivan, who finished joint-sixth and were the highest-placed players not already in the field for Carnoustie.
Fox shot 68 to Knox’s 66, with Jorge Campillo going round in 65 to finish one off the pace at 13 under.
Campillo may have managed the lowest score on Sunday, but the performance of the day surely came from Jon Rahm, who managed a six-under 66 - rounded off with four birdies in succession - despite registering a triple-bogey seven on the second.
Erik van Rooyen led by four overnight but fell away with a final-round 74. Playing partner Joakim Lagergren fared even worse, shooting 75.
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