Former champions Woods and Garcia improved by a stroke in completing their last four holes, while 25-year-old Lingmerth lost one when he bogeyed the 18th to stand on 11-under-par 205.
Lingmerth posted a three-under-par 69 for the round, while Woods shot 71 and Garcia 72.
One stroke back were Sweden's Henrik Stenson, who completed a 71, and Americans Ryan Palmer (70) and Casey Wittenberg (70).
Wittenberg, who competed last year on the Web.com Tour with Lingmerth, completed a birdie-birdie-birdie finish to burst up the leaderboard.
Two strokes off the pace was 49-year-old Jeff Maggert, who had completed his round before play was halted due to darkness following a day marred by bad weather and threat of lightning.
Tied for eighth on eight under par were Englishman David Lynn (68) and Hunter Mahan (71).
Eight players came back to TPC Sawgrass to finish up, and Garcia took an immediate step back with a bogey at the 15th, but rebounded with birdies at the par-five 16th and par-three 17th, where his tee shot glanced off the pin.
After Lingmerth bogeyed the last following a poor tee shot, Woods grabbed a share of the lead when he birdied 16 but missed birdie tries on the next two holes.
The finish put Lingmerth into a last-group pairing with Garcia, while Woods will play Sunday's final round with Wittenberg.
Long-time rivals Woods and Garcia, who have been at odds with each other over the years, had a contentious moment on Saturday.
The Spaniard felt Woods did not show him respect when the American prepared to play his shot from within the trees 40 yards away, eliciting a crowd roar when he pulled out his five-wood for a daring escape, as Garcia was making his swing from the fairway on the second hole.
"We did not talk about the situation," Garcia said when asked whether he had cleared the air with Woods since the incident, adding he was pleased at not being paired again with the world number one.
"I think it's good for both of us. We don't enjoy each other's company," said Garcia.
"You don't need to be a rocket engineer to figure that out."
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