Rai entered the final round without a top-10 finish in a major and had never placed better than 19th in 12 previous major starts.
Playing in windy conditions on Aronimink’s difficult greens, he recovered from a bogey at the eighth with a 40-foot eagle putt at the par-five ninth. He then made birdies at the 11th and 13th to become the first player of the week to reach seven-under.
Germany’s Matti Schmid briefly moved within one shot of the lead after a birdie at the 13th, but Rai responded with a tap-in birdie at the par-five 16th.
His decisive moment came at the 17th, where he curled in a long birdie putt over a ridge to move to nine-under. Schmid bogeyed the 15th soon after, leaving Rai with a four-shot cushion.
“I definitely wasn’t trying to hole that putt,” Rai said. “It was about the speed of the putt. Nice to see it go in.”
Rahm and Smalley share second
Rahm, a two-time major winner, birdied the ninth to reach five-under but was unable to make a charge on the back nine. He finished with eight pars and one bogey after the turn.
“I played really good golf,” Rahm said, adding that he struggled with the speed of the greens. “What Aaron did today, catching him could have been very difficult.”
Smalley, who began the day with a one-shot lead, slipped back after a double bogey at the sixth and a bogey at the eighth. He recovered late with an eagle-bogey-birdie finish to share second place.
Justin Thomas, Matti Schmid and Ludvig Aberg finished tied for fourth on five-under 275.
McIlroy finishes tied seventh
World number two Rory McIlroy, who won the Masters last month, shot 69 to finish tied seventh on 276 alongside Xander Schauffele and Cameron Smith.
American Kurt Kitayama produced one of the rounds of the day with a bogey-free 63, matching the lowest final round in major championship history.
Defending champion and world number one Scottie Scheffler shot 69 to finish on 278.
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