The 2023 champion saw a six-shot overnight advantage cut to just one during an engrossing major final day, where Burns threatened to produce the largest final-round comeback in US Open history and Scheffler stayed in contention to complete the career Grand Slam.
Clark stuttered to a front-nine 38 and failed to regain control of the tournament over the closing stretch, with a stunning birdie at the par-five 16th cancelled out by a three-putt bogey at the next to leave him just one ahead with one to play.
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The American missed the fairway at the par-four last but found the green with his approach, then two-putted from 50 feet to close out a final-round 73 and secure a second major victory in four years.
Clark ended the week on four under and one ahead of Burns, who started the day seven back but charged to a runner-up finish after a three-under 67, with Tom Kim finishing third on one under in front of world No 1 Scheffler, Keith Mitchell and JT Poston.
How Clark pipped Burns in major thriller
Burns - playing three groups ahead of the overnight leaders - fired his approach at the first to tap-in range and added another birdie at the third, which saw him move within four of the lead when Clark failed to get up and down to save par at the second.
Momentum continued to move Burns' way with a 25-foot birdie at the par-five fifth and another from nearly 50 feet at the eighth, as Clark holed from 15 feet to save par at the fourth but saw a chip roll back to him on his way to a bogey at the next.
Burns slipped back when he bogeyed the ninth but found himself back within one when Clark missed from four feet at the seventh, with the overnight leader then having to produce a sensational up and down to scramble a par at the ninth to stay ahead at the turn.
Clark started his back nine with a close-range birdie but bogeyed the par-four 13th, while Burns responded to missing from four feet to save par at the 15th by making a key birdie at the next to get back to three under.
Burns tentatively left his birdie attempt short at the 17th then sunk to his knees after seeing his effort at the last agonisingly shave the edge of the cup, as Clark - who received jeers from the New York crowd - holed from 25 feet at the 16th to double his lead.
Clark gave Burns hope when he three-putted from 70 feet to bogey the 17th and then put himself in three-putt territory again at the last, where he lagged his birdie effort to kick-in range before tapping in to claim the $4.5m first prize.
"It [winning] feels pretty amazing," Clark said. "I played some ugly golf the last two days but my putter and short game kind of kept me in it. To make all the putts I did has been amazing."
Kim marked his 24th birthday by posting a level-par 70 and grabbing third spot, one of just three players to end the week under par, while Scheffler mixed two birdies with three bogeys to miss out on completing the career Grand Slam on his 30th birthday.
England's Tyrrell Hatton shared seventh spot on one over with Gary Woodland, Sam Stevens and Joaquin Niemann - who carded an 11 on the par-four sixth during his opening round after a misconduct penalty, as Tommy Fleetwood, Justin Rose, Aaron Rai, John Parry, Sahith Theegala and Xander Schauffele all ended the week on two over.
Matt Fitzpatrick, the 2022 champion, faded with rounds of 74 and 73 over the weekend, while Rory McIlroy's bid for a second major of the season saw him finish in tied-32nd on six over.
What's next?
The PGA Tour heads to TPC River Highlands in Connecticut for the Travelers Championship, the latest Signature Event, while the DP World Tour is in Torino for the DS Automobiles Open d'Italia.
Both events are live from Thursday on Sky Sports Golf, along with the final men's major of the year - The Open at Royal Birkdale - from July 16-19. Get Sky Sports or stream with no contract.
