The 25-year-old Lingmerth rolled in an eight-foot eagle putt on the par-five 16th and sank a 10-foot birdie putt at the par-three 17th to reach a total of 12-under before play was called off due to failing light.
Lingmerth was four-under for the day at golf's unofficial 'fifth major', his late surge putting him two clear of overnight leader Garcia and world number one Woods, who were both playing the 15th hole when play was halted, and compatriot Stenson, who had completed 16.
Stenson, a former Players winner along with Woods and Garcia, briefly grabbed a two-stroke lead with a birdie at the ninth hole of the TPC Sawgrass's Stadium Course following a delay of nearly two hours due to the threat of lightning.
But the big Swede bogeyed the par-five 11th and par-four 15th to slip back.
Tied for fifth place on nine-under were 49-year-old Jeff Maggert, who completed a six-under round of 66, Ryan Palmer (through 16 holes) and Casey Wittenberg (17), who competed last year with Lingmerth on the second-tier Web.com Tour.
Eight players yet to finish their third rounds will return to the course to complete them from 7:10 a.m. local time (1110 GMT) on Sunday before the final round commences.
The solidly-built 5-ft-7in (1.70 metres) Lingmerth broke clear at the end of a see-sawing day that began with Garcia and Woods battling for the lead.
Garcia and Stenson both took charge at times but the younger Swede Lingmerth charged to the top of the leaderboard after putting an inconsistent front nine behind him.
"Obviously thrilled to be in this position," Lingmerth, who had missed his last five cuts on the PGA Tour, told reporters.
"It will probably be a little hard to sleep, maybe. But I'm feeling great about things so hopefully that won't be a problem.
"Might have some butterflies, but we'll see."
Garcia and Woods, whose rivalry dates back 14 years, had a lively first nine, with the Spaniard glowering toward the world number one after getting ruffled on the par-five second hole.
Noise from the gallery watching Woods distracted Garcia during his backswing on his second shot from the fairway, the crowd roaring in encouragement when the American pulled out a five-wood for his escape from the trees on the left.
Garcia sliced his shot well into the trees on the right and ended up taking a bogey, while Woods birdied the hole to leapfrog him into the lead.
The Spaniard complained that Woods should have waited until he had played his shot before preparing for his own.
"There's really not much you can do," Garcia said. "I think that I try to respect everyone as much as possible out there. I try to be careful what I do to make sure it doesn't bother the other players."
Woods claimed he was innocent of any gamesmanship, and returned serve at the Spaniard.
"The marshals, they told me he already hit, so I pulled a club and was getting ready to play my shot, and then I hear his comments afterwards, and not real surprising that he's complaining about something," said Woods.
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