The three leading players in Melbourne not already exempt earned their places at Royal Liverpool, with Lee finishing in third at eight under after a 69.
Cañizares made a charge up the leaderboard with a closing 64 that got him to seven under but Barron was three shots shy of that stood on the 17th tee.
He picked up a birdie on the penultimate hole but the best was yet to come as he rolled home a huge eagle putt on the last, sparking big celebrations from the local fans and sealing Barron an Open debut.
"I'm absolutely stoked," he said. "It’s always been the one I’ve said to everyone that I’d want to win if I had the option.
"I love the history of it. I love playing golf in Scotland and I love playing golf in the UK and through Europe and that’s always been where I’ve wanted to kind of set my craft.
"I won a regional, the first qualifier at Little Aston I think in 2018, but I was unsuccessful in final.
"I’ve had a few mates kind of get through to qualifying the last few years and know that I’m probably pretty close. I feel like the guys that have got through are pretty similar players to me. That’s kind of inspiring a little bit, but just can’t wait. It hasn’t sunk in yet."
Lee will be making his third consecutive appearance in the oldest Major Championship, while Cañizares will be making his fourth appearance and first in a decade.