It was also the lowest score in Woods' 66 career trips around the Stadium course, where he won in 2001 and 2013.
Woods made the cut with nothing to spare after late mistakes from Jordan Spieth and Justin Thomas on Friday evening pushed the cut back to one under, and he took full advantage as he fired a superb 65.
Three of the world's top-10 in Rickie Fowler, Rory McIlroy and Hideki Matsuyama, along with five-time major victor Phil Mickelson, found themselves with the weekend off after missing the cut in the Players Championship on Friday. "15 under through two rounds, any two rounds out of four that you can get to 15 under adding together is incredible, let alone the first 36".
Simpson then came within inches of holing out from the fairway at the fourth and, although he missed the green at the short eighth and failed to get up-and-down, he got the shot back immediately with a nice pitch and putt for a four at the long ninth.
"I think it's a lovely golf course in the sense that I could go shoot a few under tomorrow because there's birdie holes out there", he said. "17's a sand wedge in there, but I'll take it". Woods surged up the leaderboard with a seven-under-par 65 during round three.
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"I got to shoot something in the probably mid-60s both days to get myself up there to have a chance or something", Woods said.
On the 558-yard par-5 11th hole, Fleetwood reached the green in 3 and sunk a sub 1-foot putt for birdie.
"65 was actually about the highest I could have shot today", Woods said after the round.
Simpson safely negotiated the final hole with a par as he signed for a 68, moments after Lee had closed on 12 under after a round containing just two birdies and 16 pars.
Tiger Woods said it felt great to shut things down after Augusta National.
"I mean to be eight-under there through 12, realistically I probably could have got a couple more out of it and got to 10 for the day", said Woods. "Iron play is good, and also putting, hitting a lot of good putts". "It's definitely gettable. I know there's a lot of pressure coming down the back nine here, but I think these guys, the way they have set up the golf course today, it's set up for these guys to go low". "We were like, 'Man, I hope he shoots 59.' And at the same time, we're saying if he shoots 59, can you imagine how hard the setup is going to be tomorrow?"