Burns went out early from the 10th tee and covered the punishing Oakmont layout with a sparkling five-under-par 65 that put him at three under on the week for the year's third major and one shot clear of Spaun (72).
Burns, who mixed six birdies with a lone bogey, ran into trouble at his final hole where an errant tee shot forced him to take a penalty drop but left his next shot on the green and then curled in a 22-foot, left-to-right putt up the hill to save par.
"Then that putt was, I don't know, six feet of break. Yeah, it was a nice one to make for sure," said Burns.
Spaun had led Burns, who finished hours earlier, by one shot with four holes to play but fell back after a three-putt bogey at the 15th, where his birdie attempt raced by the cup, and another dropped shot at the par-three 16th.
A determined Spaun drew level with a 22-foot bounce-back birdie at the 17th but was unable to scramble for par at the last after finding the rough behind the green.
Viktor Hovland chipped in from the greenside rough for eagle at his eighth hole, the par-four 17th, en route to a 68 and was two shots back of Burns and one of just three players in the clubhouse under par on the week.
Pre-tournament favourite Scottie Scheffler, fresh off the worst first-round score relative to par of his majors career, mixed four birdies and five bogeys for a one-over 71 that left him at four over on the week.
"Today was, I think, with the way I was hitting it, easily a day I could have been going home and battled pretty hard to stay in there," said Scheffler.
"I'm four-over. We'll see what the lead is after today, but around this golf course I don't think by any means I'm out of the tournament."
McIlroy, who has been struggling to regain his form ever since completing the career Grand Slam at the Masters, double-bogeyed two of his first three holes and was sitting right on the projected cut line of seven over late in his round.
He was also trending during the day after tossing his club and smashing one of the tee box markers on the 17th after sending his ball straight into a bunker.
Defending champion Bryson DeChambeau, who came into the week looking to become the first back-to-back U.S. Open winner since Brooks Koepka in 2018, was at 10 over on the week with one hole to play.