Kuchar wobbled to a 1-over 36 on the front nine and let a whole host of players back into the festivities before making three straight 3s on the back nine and two birdies in his first three holes.
"To shoot seven under back-to-back is unexpected, but awfully excited", Kuchar said.
American Kuchar - who is chasing his first win since November - remains in control following his four-under-par 66 at Wai'alae Country Club in Honolulu.
Biggest disappointment: It was a rough afternoon for Keith Mitchell, who turned a spot in the final group into a tie for 16th.
Kuchar had a 54-hole score of 193 in Mexico, the lowest of his career.
He'll go into Sunday's final round at 18-under 192 with a two-shot edge on Andrew Putnam, whose strong start allowed him to tie for the lead early in the third round. Putnam missed an 8-foot birdie attempt on No. 18. Five 3s in your first seven holes on the back nine with the tournament in the balance is a nice way to close a two-week stay in Hawaii.
"It was not at all what I was looking for", Kuchar told Golf Channel of his start.
"I feel like my game has turned into a game that can play well at any given course", Kuchar said.
The PGA Tour only began keeping hole-by-hole records in 1983, and no one had ever made three eagles in one round on par 4s since then. He holed out for eagle three times from the fairway - a sand wedge from 101 yards on No. 10 at the start of his round; a 9-iron from 149 yards on No. 16, and a gap wedge from 135 yards on No. 6. Reavie made more eagles - three - than birdies, of which he had only one.
I get where he's coming from and know that he's just trying to build confidence, but I might have to amend my "Spieth will win three times in 2019" take in a few weeks.
After shooting 67 on Thursday, Leishman fired a six-under-par 64 (131) on Saturday (AEST) to be alongside American Ted Potter, five strokes behind Matt Kuchar, who dazzled with back-to-back 63s for a 14-under-par two-round total of 126.
Bryson DeChambeau tied his career low round of 63.
Also tied for fifth were Charles Howell III and 54-year-old Davis Love III, who had one of his better putting rounds. "I have to stay aggressive".