Scott (68) briefly reached eight under with a sensational eagle at 15, only to three-putt for bogey at the next hole, while Koepka - one of two overnight leaders along with Bryson DeChambeau - recovered impressively from a shaky start and birdied the last for a 71.
The 29-year-old was eight shots off the lead as he signed his card, but low scores were the order of the day and the four-time major victor was certain to be too far back to mount a challenge on Sunday. He had four birdies and added an eagle on No. 8 on the front nine alone.
Dustin Johnson (70) and Justin Harding (69) were a shot behind at 6-under, alongside four-time Masters victor Woods, who is sitting ominously just behind the leaders as he hunts a first major since the 2008 US Open.
But their feats were all eclipsed by Woods, who signaled he would be a threat to capture his first major title since the 2008 US Open and first Masters title since 2005 - astonishing after years of knee and back injuries that made him doubt if he would ever play competitively again. When he mounted the 18th tee here 28 years ago, he picked out his Persimmon driver and smashed his tee-shot clean over the left-hand bunker.
The expected fireworks started early as one of the USA hopes Finau provided the main plot twist, racing to the top of leaderboard with a sparkling six-under 30 front nine featuring four birdies and an eagle.
He didn't show any trouble the rest of the way, and didn't find much interest in talking about it after his round except to say "accidents happen" and "it's all good".
At the par-5 15th, Woods curled in a 30-footer for birdie, punctuated with a right fist pump to put himself one stroke off the lead.More news: It's deadline day for Trump's tax returns
Webb Simpson, who moved into contention after shooting an 8-under 64 that leaves him at 9 under, is hoping for weather to make an impact.
With the forecast calling for heavy rain and thunderstorms late on Sunday officials have chose to send golfers off in threesomes from both the first and 10th tees rather than traditional pairs in an attempt to get the final round completed and avoid a Monday finish to the year's first major.
It worked wonders as he appeared to be in considerably less pain while putting together a round that was almost flawless, although Day said his wife Ellie deserved just as much credit for his remarkable recovery.
Scott returned from a 40-minute weather delay and fired a 231-yard second shot into the par-5 15th hole.
McIlroy was also less than happy with his tee shot on the par-three fourth, but holed from 32 feet for an unlikely birdie, only to then drop shots on the sixth, seventh and ninth, where his approach plugged in the top of a greenside bunker. "I'm in a good spot to kind of run with them".
Finau, a 29-year-old American became the third player to shoot an 8-under 64 in the third round of the Masters.