Soggy conditions changed the dynamic of the course but not the difficulty as Burns and Spaun traded blows in the final group.
Both are winners on the PGA Tour but they also share play-off defeats this season – Spaun at The Players Championship by Rory McIlroy in March and Burns only last week on the fourth play-off hole as Ryan Fox won the Canadian Open.
The two briefly swapped places at the beginning of the round but were level for the majority of Saturday and both picked up birdies on the 17th to move clear of Scott on four under.
But Spaun, who still boasts the only bogey-free round of the week on Thursday, hit an erratic tee shot at the 18th. That led to a bogey and cost him a place in Sunday's final group.
That spot alongside Burns will be filled by Scott, who is making his 96th consecutive appearance at a major championship.
His second shot at the 14th was sublime, landing a foot from the pin, and his fist-pump celebration to raucous cheers after making birdie on 17 showed his heart – and popularity – remain undimmed.
On Friday, Scott said a second major title would "go a long way" in fulfilling himself. He is well placed to make that happen on Sunday and has the added bonus of being the only player in contention to have contested the 2007 and 2016 US Opens at Oakmont.
"I was less overwhelmed coming to Oakmont this time, and that's not a knock on the golf course, but maybe just a couple trips around the U.S. Open here, I knew what to expect," said Scott, who missed the cut in 2007 and finished joint 18th in 2016.
And do not discount Hovland, who despite coming within inches of hitting his opening drive out of bounds, retained his composure to produce a typically swashbuckling round that featured three birdies and three bogeys.
"I'm well aware that I've got a chance, and if I shoot a low round of golf then anything can happen," he said of his chances of winning a first major on Sunday.