The projected cut was at two under numerous times, but it dipped to one under late in the day to let in Woods, world No 2 Thomas, and Spieth, who staged a valiant fightback from his opening 75 to card a seven-birdie 68.
Simpson is soaring at 15-under and has established a five-shot lead at TPC Sawgrass.
Meanwhile, Simpson tied Greg Norman's 54-hole scoring record, as the American remained in cruise control atop the on day three.
Woods wrapped up his round with a par on No. 17, after just missing a 30-foot birdie putt on the 126-yard par 3, and then found himself with a 15-foot birdie attempt on No. 18.
The 2012 US Open champion drained three from close to 20 feet on 12, 13 and 14 before holing out from off the green at the next, his ball hitting the cup at pace and dropping.
"I was pretty satisfied", Simpson told Todd Lewis of Golf Channel.
Simpson, whose unique putting action is described by his caddie as the (Matt) Kuchar-claw, has converted 39 of 41 putts from 10 feet and in, and punctuated his round with a17-footer for par at the last hole.
Several players are both 9-under and 8-under, including Jason Dufner, Jimmy Walker, Jason Day, Jordan Spieth and Tiger Woods. The highlight of his day was when he holed a bunker shot on the par-five 11th, giving him eagle.
Woods initially continued that form on the way home and was joint-fifth at eight under through 12 after consecutive birdies brought the nine-under 63 course record into view.
He answered questions from the media in a press conference after his round, per the Golf Channel broadcast. As the wind picked up later in the day, scores stayed relatively high, making Simpson's round that much more impressive. "I should be hitting them inside flag length and didn't do that at all today". It's just a matter of playing and executing.
"I just need to focus on the good progress I made and try and take it into the weekend".
Spieth finished strongly with three successive birdies at the end of his third round. He birdied 16, 17 and 18. But 17 is a hard hole to get through. "I don't think any lead is safe". "I think that some of the guys will probably shoot better scores than I did out there today". - Sometimes, it is as simple as hitting a nice drive, seeing the ball go where you want it to go and getting that first putt to drop. - If Webb Simpson was feeling any pressure by Tiger Woods leap up the leaderboard on Saturday morning it certainly wasn't reflected in his play.
Here's a link to all the tee times for Sunday's final round. Simpson shot 4-under 68 to climb to 19 under and take a seven-shot lead on the field.
And with Mother's Day falling on the final day Simpson revealed he has an added extra incentive to win.
Look for Johnson to have a strong finishing round.
Simpson won his United States Open title anchoring a belly putter into his midsection.
Fans should get a thrill out of watching Woods and Spieth play together Sunday.
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