Woodland is a three-time victor on the PGA Tour but has never won a major championship. Spieth didn't get under par for the tournament until his seventh hole Friday - the par-3 16th hole - and he managed to do enough right for a 66 to get within seven shots of the lead.
Kisner shot 29 on the back nine - his first nine - and was threatening to shoot a tournament-record 62 but bogeyed the final hole and settled for 64. "The key is to get the ball in the fairway and attack from there". Time after time he put himself in position to go lower, and time after time he let the opportunity slide by.
Ten players were within four shots of the lead, which includes defending champion Justin Thomas (68), Jason Day (67) and Stewart Cink, the 2009 The Open Championship victor who played with Woods and matched his 66.
US Open champion Koepka looked set to carve out a big advantage in his bid for a third major in two years, but came unstuck with successive bogeys on the back nine before recovering with a birdie on the 17th to close with a 66 of his own and 12-under 198.
The 14-time major champion, in the eighth month of his comeback from spinal fusion surgery after years of back injuries, shared third on eight-under-par after seven holes at Bellerive Country Club.
Rounding out the Australian contingent is Marc Leishman (72) and Cameron Smith (73), who both tumbled down the leader board to be one over and three over respectively.More news: Derby County make Malone their eighth summer signing
More news: Why the Galaxy Note 9 is the new king of the enterprise
More news: Stan Kroenke makes share bid for full control of Arsenal
His nearest challenger is former Masters victor Adam Scott of Australia, who carded a fine five-under 65 for 10-under 200, two shots back. "I played better than my score indicates", said Woods, who had to finish off his second round on Saturday morning after storms curtailed Friday's play. "So I just got to be a little more aware of and make the adjustments better than I did today".
337: Koepka's average driving distance in the third round. "Having said that, I would have shot a much higher score yesterday".
Woods moved inside the top-10 with five birdies in his first eight holes, as Rory McIlroy slipped further out of contention with an erratic one-over 71.
"I ran it by", Woods said.
Most notably, the world number 50 missed a 20-foot putt for eagle on the par-five 17th and then saw his four-foot birdie attempt slide by. "And that might mean shooting 5 under again to hang around, but I would love to be in the mix coming down the stretch and have the chance to hole some putts to win". Spieth dropped for his second provisional and punched out. But I've given myself a shot at winning and I'm excited about that. The kicker came at the par 5 17th. I feel like I played well [Saturday].