He carded a 66 for a 16-under 200 total, matching the best 54-hole score at Bay Hill, while American Morgan Hoffman (71) is alone in second place on 14-under.
Four players are three shots behind, including Australian Matt Jones, who birdied the final four holes, and defending champion Matt Every.
While Stenson surged, world number one Rory McIlroy backtracked with a 71 to fall seven shots off the pace after running up three consecutive bogeys from the 14th hole.
“I just hit some great shots and some great putts,” Stenson told NBC. “Of course you’re picking up pace if you go four-under two days in a row on the closing stretch.”
Stenson, 38, is no stranger to victory, with four wins on the PGA Tour and eight on the European Tour.
His form this week is a continuation of his recent strong play. He has finished fourth the past two weeks.
“I’ve got to keep the same mindset I’ve been keeping these last 10 or 12 rounds,” he said.
“It’s been working pretty well so far so I’m going to try more of the same. It’s a new day tomorrow. Whatever has happened these first three days has put me in a nice position but it’s got to be done all over tomorrow.”
Second-placed Hoffman lamented over some perceived bad luck but did enough to remain in contention.
“It was a weird day,” the American said. “I got some squirrelly lies in the fairway, some mud balls, and up against some lips in the bunkers. Steve, my caddie, was really good at keeping me under control.”
McIlroy, meanwhile, made a strong move before an error-strewn three-hole stretch all but ended the Northern Irishman’s hopes of victory.
A bogey at the easy par-five 16th was particularly disappointing.
(Reporting by Andrew Both in Cary, North Carolina, editing by Gene Cherry)
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