Burns, whose most recent win came at the 2023 WGC Match Play, stood on four-under 206 after 54 holes while Spaun, whose only PGA Tour win came at the 2022 Texas Open, shot 69 with a bogey at 18 to share second on 207.
Scott, the 2013 Masters champion, birdied three of the last six holes to fire a 67 and reach 207, two strokes ahead of Norway’s Viktor Hovland in fourth.
The four will play in the last two pairings once again on Sunday in a tension-packed final round for the US$4.3 million top prize.
Last-group playing partners Burns and Spaun traded the lead all day, a wild early exchange ending when Burns made a tap-in birdie at the fifth to match Spaun at three-under.
Burns sank a seven-foot birdie putt at the par-three 13th to seize the solo lead at four-under but fell back with a bogey at 16.
Both birdied the 17th, Burns landing his approach inches from the hole and Spaun sinking a putt from just inside 14 feet.
At 18, both went into the right rough but Burns reached the green while Spaun found a bunker. Spaun blasted out and two-putted for bogey while Burns sank a four-foot par putt to grab the solo lead.
Scott, 44, can be the second-oldest US Open winner after American Hale Irwin, who was 45 when he won in 1990.
Scott landed his tee shot inside five feet at the par-three 13th and dropped his approach inches from the hole at 14 to set up birdies, then escaped a bunker at 17 and sank a 14-foot birdie putt.
Hovland blasted out of a bunker to set up a tap-in bogey at 17, but made bogey from a bunker at 18.
Mexico’s Carlos Ortiz fired a 67 to stand fifth on level par 210, a stumble at the last hole dooming a bogey-free bid.
Ortiz, a LIV Golf player who won an Asian Tour event in Macau in March, made the cut for only the third time in 10 major starts and has never finished a major in the top 50.
“It would be great. Winning any kind of tournament is awesome,” Ortiz said of capturing the title. “I’m just going to do my best and hopefully that’s enough. If not, that’s all I had.”
England’s Tyrrell Hatton, another LIV player, birdied three of the first four holes on the back nine and shot 68 to stand on 211, sharing sixth with South African Thriston Lawrence.
Big names struggled with world number two Rory McIlroy making bogeys on three of his last five holes to fire a 74 and stand on 220.
“I was hoping to play better, but I didn’t,” McIlroy said, calling his tournament “pretty average” so far.
“The name of the game this week is staying patient,” McIlroy said. “But it’s one of those golf courses that you can lose patience on pretty quickly.”
Northern Ireland’s McIlroy, who completed a career Grand Slam by winning the Masters for his fifth major title, had bogeys on the third and ninth holes, followed a birdie at 10 with bogeys at 11, 14 and 16 and closed birdie-bogey.
Top-ranked Scottie Scheffler, seeking back-to-back major titles after taking last month’s PGA Championship, fired a 70 with four birdies and four bogeys, sinking a five-foot birdie putt at 17 to stand on 214 after 54 holes.
“I had a hard time getting momentum,” Scheffler said. “I made a few silly mistakes, a couple three-putts, which is just a killer.”
Two-time major winner Jon Rahm of Spain made a double bogey at 15 and found a bunker on his way to a bogey at 18 to fire a 73 and stand on 217.
Third-ranked American Xander Schauffele, the reigning British Open champion, shot 71 to stand on 217. (AFP)