The Swede, who turned professional in June 2023 and has two Ryder Cups and three tournament victories to his name, improved his two-shot halfway lead by one after a round of 71 on Saturday to get to 13 under.
Two-time major champion Xander Schauffele dropped back after a 74, Cameron Young (72) was within three until he double-bogeyed the last and Justin Thomas and Corey Conners could only manage level-par rounds of 72 to close on eight under.
Robert MacIntyre's superb seven-under round of 65 - his best at Sawgrass - had nine birdies, with a brilliant finish of three in succession from the 15th and concluding with a nerveless 10-foot par putt to get to seven under.
Aberg would have had a bigger advantage were it not for his second bogey of the day at the 18th, having missed a short birdie putt on 17 to open up what would have been a five-shot lead as Young was hitting it in the water down the last.
"Definitely would have loved to come out of 16 and 17 with at least one birdie, and then obviously the three-putt on 18 kind of stings, annoys me a little bit," Aberg said.
"But overall, I started the day with a two-shot lead and ended with a three, so that's a positive."
It left Michael Thorbjornsen as his nearest challenger after a 67 moved him to 10 under.
Aberg's Ryder Cup teammate Matt Fitzpatrick (69) was left to rue "sloppy errors" which left him six shots back in a tie for fourth.
"You can't afford any mistakes. I've had four dropped shots which you can't really afford if you are trying to win the tournament," Fitzpatrick told Sky Sports.
Jason Day (72) and Min Woo Lee (70) are the best-placed Australians in a tie for 26th at four under, while Adam Scott (71) is one under.
Rory McIlroy continued to struggle as his slim hopes of a title defence ended after a level-par 72.
Two of his three bogeys came after finding water off the ninth and 12th tees, the second cutting short his momentum after successive birdies threatened to kickstart his back nine to leave him one over.
With AAP.
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