Burns looked plenty comfortable as he made his way around the treacherous Oakmont layout, never panicking when he got into trouble, and carded a one-under-par 69 that brought him to four under on the week.
Scott, the 2013 Masters champion who is playing his 96th consecutive major, missed several birdie putts but converted three on the back nine for 67, equalling the low round of the day and will play in Sunday's final pairing with Burns.
Spaun, who began the day one shot back of playing partner Burns, had a share of the lead late but bogeyed the par-four 18th where he put his second shot in a bunker and was unable to convert a 21-foot par putt.
World number 14 Viktor Hovland (70) was alone in fourth place and three shots off the pace.
Hovland birdied the penultimate hole after a stunning chip from the thick rough that settled inside three feet, but gave it right back at the last, where he made bogey after another errant tee shot.
Meanwhile, the third round was another challenging day for Kiwi Ryan Fox.
He shot a three-over 73 for the second straight day, which leaves him at eight over for the tournament.
It didn't take long for the renowned Oakmont Country Club to bite Fox, with a double bogey at the third, following pars at the first two holes.
His tee shot on the par-three sixth, in particular, was exceptional, as he hit the ball to within a foot of the hole, but that was as good as it got for Fox, as he dropped a shot on the seventh and then posted a second double bogey of the round at the par-four ninth hole.
He steadied on the back nine, dropping just the one shot, when he bogeyed the 15th hole.
LIV Golf's Carlos Ortiz was enjoying a rather stress-free trip around the notoriously treacherous Oakmont layout until a closing bogey but still signed for a 67 that left him at even par on the week and alone in fifth place.
Fellow LIV player Tyrell Hatton (68) was a further shot back in sixth place with South African Thriston Lawrence (70).
Pre-tournament favourite Scottie Scheffler was unable to make a big move as he mixed four bogeys with four birdies for an even-par 70 that left him at four over.
"I felt like I had a hard time getting momentum," said Scheffler. "I made a few silly mistakes out there, a couple three-putts, which is just a killer."
Rory McIlroy, trying to shake off a Masters hangover since completing the career Grand Slam at Augusta in April, started the day nine shots off the lead and carded a four-over-par 74 that left him at 10 over.
"The name of the game this week is staying patient and try to do a good job of it out there, but it's one of those golf courses that you can lose patience on pretty quickly," McIlroy said. "I was hoping to play better but I didn't."
