In second place on three-under after back-to-back birdies at six and seven was South Korean Kim Si-woo through 16 holes.
World No 1 Scottie Scheffler began his first round only minutes after Spaun reached the clubhouse, the three-time major winner favoured even as Oakmont, while vulnerable, had danger lurking on every hole.
World No 2 Rory McIlroy finished with a four-over 74 after dropping four shots on his last four holes.
Defending champion Bryson DeChambeau carded a three-over 73, with just two birdies.
Spaun, who won his only PGA Tour title at the 2022 Texas Open, lost a playoff to McIlroy at this year’s Players Championship.
Spaun sank a 20-footer for birdie at 10 to start and dropped his third shot inside three feet at the par-five 12th to set up a birdie.
He also birdied the par-three 16th and escaped the rough at 17 to sink an 11-foot putt and reach four-under.
Spaun, never in the top 20 in eight prior major starts, parred 18 to shoot 31, the best nine-hole start ever in a US Open at Oakmont.
After finding the Church Pew bunkers left of the fairway at the par-five fourth hole, Spaun sank a testing putt to save par.
He also found a bunker at the par-three sixth but sank another lengthy par putt.
“Those are huge, especially at a US Open, to keep your round going and not go the other way,” Spaun said.
McIlroy, who completed a career Grand Slam by winning the Masters in April, had back-to-back early back-nine birdies, sinking a 27-foot putt at 11 and reaching the green in two at 12 to set up another birdie.
But McIlroy stumbled with a three-putt bogey at the first.
He needed three shots to escape the right rough but sank a 30-foot downhill putt to salvage a bogey at the par-five fourth.
Two bogeys and a double bogey followed.
Phil Mickelson, who turns 55 on Monday, was four-over through 15 in his bid to complete a career Grand Slam after six US Open runner-up finishes.
Kiwi Ryan Fox began with a birdie at the second hole but followed it up with a bogey to be even through three holes.
Lowry, Moldovan magic
Ireland’s Shane Lowry holed out from the fairway from 160 yards for the first US Open eagle at Oakmont’s third hole, but he finished nine-over.
American Maxwell Moldovan made the first US Open eagle at the first, holing out from the fairway from 189 yards, then looking to the heavens with a smile. He fired a 76.
Scheffler is playing alongside US compatriot Collin Morikawa, a two-time major winner ranked fourth in the world, and Norway’s Viktor Hovland.
Scheffler, whose nine wins last year included Paris Olympic gold, has won three of his past four starts.
After capturing last month’s PGA Championship for his third major crown, two-time Masters winner Scheffler is trying to become the first man to win consecutive majors since Jordan Spieth in 2015.
