Dustin Johnson at least improved his chances of staying at No. 1 with four birdies over his final 10 holes for a 69.
Simpson, who has not tasted success for almost four years, immediately declared his intentions on day three with a birdie at the first and then another at the par-4 fourth to get to 17 under.
It was really important that I didn't lose my head because every hole is hard around here.
Tiger Woods will be sticking around for the weekend's action but only just scraped by the one-under cut-line, after rounds of 72 and 71, along with pre-tournament favourites Jordan Spieth and Justin Thomas. He's just four strokes back of second, but 11 back of Simpson, who carded a 4-under 68 on Saturday.
The North Carolina native, and victor of the U.S. Open in 2012, had 248-yards to the flag at 11 but sent his second shot some 33-yards past the hole and into a rear greenside bunker.
Schwartzel didn't make a single bogey on Day 2 at TPC Sawgrass and is now ninth in the field in strokes gained: tee to green.
It sent Simpson to a remarkable 19-under par and then seven shots clear of his rivals.
The 14-time major victor, fit again after an injury-enforced spell off the tour, had struggled off the tee over the first two days and did not have too many realistic birdie chances.
"[This round prepared me] a good amount", Simpson added. "I'm hungry to get back in the winner's circle", he said after having tied Greg Norman (1994) for the best third-round score. To be 11-under par, it's a shame he doesn't finish it off.
Two dead, four injured in Paris knife attack
Two of the wounded in the attack are in a serious condition but do not have life-threatening injuries. Islamic State claimed responsibility for Saturday's attack, but did not provide proof for its claim.
"My dad used to always look after me", Schwartzel said. "After what happened yesterday on 17, I wanted to play a good solid hole and I did that".
"I'm telling you, if you don't miss a putt inside of 15 feet, golf is pretty easy", Spieth. "But I'm trying not to dwell there".
"So I'm just looking forward to getting some good dinner and some rest and coming out and enjoying it tomorrow".
Woods and Spieth will play alongside each other over the final round and with Woods delighted after his lowest score by three shots this year.
"I'm going to just keep my hat down and play my game", Lee said.
Simpson was between a hard gap wedge and an easy pitching wedge, and after a long conversation with Tesori, made a decision to go with the gap wedge, aiming for a spot 138 yards away, up the slope of the green, short of the back-center hole placement. Woods finished on minus eight, one behind clubhouse leader Jason Duffner.
"Overall on the whole day I just hit a lot of quality shots and a 65 was probably the highest I could have shot". "Not to be negative, but I think Martin Kaymer had a 10-shot lead in Abu Dhabi and didn't win [in 2015], so you can never be completely comfortable - especially on this course". He made his birdie putt, on his way to a record-tying 19 under through three rounds.
Simpson's struggles at the 17th boiled down to being unsure of about hitting a pitching wedge or sand wedge from 140 yards.