Finau, who dislocated his ankle while celebrating an ace in last year's Par-3 Contest on the eve of his Masters debut, surged into contention with an eight-under 64 that brought him to 11 under, two shots off leader Francesco Molinari.
His also tied the tournament's front-nine record with a six-under 30.
"I can attack the golf course if I'm hitting my driver well, and I have been thus far. I felt good," said Finau. "Going into today, I felt calm and comfortable, and I think my score showed that."
While Finau courageously finished in a share of 10th place at last year's Masters, he was still largely known as the golfer who blew out his ankle while celebrating a hole-in-one.
Finau even had some fun at his own expense this week, sending the gallery into fits of laughter when he put on a hilariously high-topped shoe featuring added support for his since-healed ankle before teeing off in the Par-3 Contest.
Finau, who started the day four shots back of the co-leaders, got off to a scorching start on Saturday, opening with three consecutive birdies.
From there he was off to the races, adding another birdie at the sixth before making eagle at the par-five eighth after his approach settled eight inches from the cup.
Finau, the first golfer of Tongan and American Samoan descent to play on the PGA Tour, added another two birdies on the back nine.
The 29-year-old American will play in the final threesome on Sunday with Molinari (66) and Tiger Woods (67).
With heavy rain and thunderstorms forecast for late on Sunday, officials have decided to send golfers off in threesomes from both the first and 10th tees rather than the traditional pairs in an attempt to get the final round completed.
Finau is no stranger to being in contention at one of golf's blue-riband events having been in the mix on the final day of last year's U.S. Open, a learning experience he hopes will pay off of Sunday.
"It's going to help because I've been in that ‑‑ I've been in that situation before," said Finau. "Last year I had the lead going into the final round of the U.S. Open, and I took a run at it towards the end.
"I feel like I've learned some things about that finish that I can apply to tomorrow, and hopefully do better."
(Editing by Peter Rutherford)
