World number two Spieth, the highest ranked player at the Waialae Country Club, overcame a rocky start and was inside the top five when he reached his penultimate hole, the par-four eighth, where disaster struck and he fell back to even par.
Overnight co-leaders Chris Kirk and Zach Johnson were in a group sharing second place on 10-under-par 130.
Joining Johnson in a tie for second at 10-under 130 are Chris Kirk (67) - who began the day with an eagle at the par-4 10th and ended it with a bogey at the par-5 ninth - Tom Hoge (65) and journeyman John Peterson (64).
Except for the tropical warmth, the two golf courses on the Hawaii swing are nothing alike.
Harman, a two-time PGA Tour victor, says his game has reached a new level since he switched to a new ball in May. "I've always kind of felt like as long as there's fairways and green and holes to putt it, then I'm going to be fine".
"I'm making putts, but I'm also putting myself in position to make those putts", he said. "I was really disappointed with the way I finished the year and the Playoffs because I wanted very badly to make that team and I didn't get it done". Defending champion Justin Thomas posted a second straight 67 to trail by seven. I'm not making everything I'm looking at, but I'm hitting a lot of good putts and made a few.
Jordan Spieth made the longest putt of his PGA Tour career - just over 90 feet on No. 5 - but didn't give himself many good looks.
But things were about to take a turn for the worse as he pulled his tee shot left and hit a tree just 220 metres off the tee. He hit a tree with his next two shots before finally finding a greenside bunker. I feel like I was a better ball striker when I was a kid than a pro.
Harman entered the final round of the Sentry Tournament of Champions paired with eventual victor Dustin Johnson.
"Missing out on the Presidents Cup hurt".