J.J. Spaun tamed the “beast” that is Oakmont and is now soaking it all in, just days after winning the U.S. Open. Yet, the 34-year-old California native tells PEOPLE on Tuesday, June 17 that he never actually believed he could win the championship. “You have aspirations and dreams of accomplishing things in this game, but my journey has always been I accomplish as I go,” Spaun says. He continues, “I’m kind of not like a huge goal-setter to where I want to check off things season-wise, tour championship, blah, blah, blah. I just kind of take what it gives me. And I never thought I would be hoisting the U.S. Open trophy, and it's still a surreal moment.” However, the father of two did just that on Father’s Day, surviving the Oakmont Country Club, one of the most difficult courses on the tour, amid dreary weather and wicked roughs to beat out a field of the world’s best. And he did so by clinching the major in epic fashion with a 64-plus-foot putt. Never miss a story — sign up for PEOPLE's free daily newsletter to stay up-to-date on the best of what PEOPLE has to offer​​, from celebrity news to compelling human interest stories. After his playing partner, Viktor Hovland, hit a shot close by and then went before him to putt, Spaun was able to get a better read on where his own ball could go. “It was just one of those things that when you win golf tournaments, you need fortunate breaks, things to go your way in order to win,” Spaun tells PEOPLE about his putt. “And that was one of the things that helped me pull the win off on Sunday.” Another break came when he had to take a break — rain moved in and delayed play for 90 minutes. All of which was ideal timing for Spaun, whose start on Sunday was less than ideal when he carded five bogeys on his first six holes. “It was huge having the delay,” Spaun tells PEOPLE. “It was actually a blessing in disguise. I kind of liken it to getting to have time in a big NBA game or any other sport where you can recollect your thoughts and come back with a game plan and assess, and most importantly just reset.” Earlier this season in March, nearly the same thing happened to Spaun when weather delayed The Players Championship, and it forced the golfer to regroup. He eventually went on to turn his round around and make the playoff with Rory McIlroy, who ended up winning the tournament. “I've had a lot of close calls this season that started to lead me in believing that I can accomplish pretty big things,” Spaun says. On Sunday, he certainly did that. The PEOPLE App is now available in the Apple App Store! Download it now for the most binge-worthy celeb content, exclusive video clips, astrology updates and more! Now a two-time PGA Tour winner, Spaun says he is relishing all that comes with the newfound notoriety — and adds that even he can’t help but rewatch that putt on 18. “I mean, I can't hide from it, it's everywhere right now,” he says. “And it was such a great moment to kind of relive and see it and just think that shot is going to be in the archives forever.” Spaun adds, “You can only dream of yourself doing that. There's so many iconic moments throughout championship history that you're like, ‘Oh man, that's awesome. Will that ever happen to me?’ And you never know what the future holds. And sure enough, I had my moment.”