Spieth, 20, carded a four-under-par 69, as did defending champion Johnson, to sit 14-under overall alongside Simpson, who had four birdies on the back nine of his 68.
However, overnight leader Zach Johnson could not consolidate the good work he had produced over the previous two days at the Kapalua Resort as he let a three-shot lead slip. Two bogeys after the turn saw him manage just a 74 to slip to fourth at 12-under.
Dustin Johnson was heading nowhere at one over for his round until an eagle at the ninth turned his form around, and he picked up a further three shots on the back nine.
Johnson told the Golf Channel: “I’ve got to take advantage of the par-fives. Today, I made a birdie and an eagle but I parred the two on the back nine which is poor for me, especially (after) hitting good drives.”
Simpson on song
Spieth and Simpson, meanwhile, each a carded a single bogey against five and six birdies respectively.
Simpson was particularly pleased with his efforts, stating afterwards: “That was really good. I feel like I am getting a little more comfortable with what I am working each day… and I was able to capitalise on those holes where you need to make birdie.
“I’ve just got to stay patient (on Monday). If you don’t get it going early, you’ve got plenty of opportunities on the back (nine).”
Behind Zach Johnson was a group of four on 11-under headed by Jason Dufner and Brandt Snedeker, while Masters champion Adam Scott was the leading non-American at 10-under.
Scotland’s Martin Laird slipped a couple of places to 21st despite a 70, while Brian Gay had the round of the day with an eight-under 65.
At 12-over, Derek Ernst was the only man of the 30-strong field to be over par for the tournament.