Finau shot a 4-under 66 that bordered on manic.
That left Woods five strokes off the pace of USA compatriot Koepka, who birdied five of his first nine holes to make the turn on 13-under par, three shots ahead of Americans Rickie Fowler and Gary Woodland with Australian Adam Scott, the 2013 Masters champion, and American Kevin Kisner another stroke adrift.
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.
Woods has been pressing to play for the US Ryder Cup team but must win the PGA to automatically qualify for the squad going to France next month to defend the trophy against Europe.
Playing alongside Woods, Rory McIlroy also birdied the eighth after getting up and down from a greenside bunker, his first birdie since the fourth hole of his first round on Thursday.
It is only the second time in major-championship history that two scores of 63 or better have been shot in the same round.
However, he could not sustain his momentum on the back nine.
"I can really tune in in the majors and I have no idea why", he said. He was at 3-over par for the second round through nine holes and 4 over for the tournament, putting him in a tie for 111th.
Scott's countryman Jason Day was flirting with a leaderboard charge on moving day, snaring two birdies on his front nine, before a dropped shot at the 10th and and a birdie at the 13th have left him at seven under.
"So to then birdie the last and play those holes in three under and six under overall was nice".
With a heat index north of triple digits and the kind of humidity that makes the championship's relocation to the more comfortable confines of May starting next year a welcome reprieve, the strongest have risen to the top.
The kinds of excited whispers typically reserved for those peering through the large crowds for a glance at Woods on the course were being used to single out one among those very crowds Saturday.
"We have to be aggressive, pars just won't be enough".
The 28-year-old, who is ranked 75th in the world, watched his tee shot pitch on the front fringe of the green before rolling in to huge roars from the galleries which could be heard around the course.
It moved him to five under on his first weekend appearance at a major after making the cut.
"I know that I can do it", Rahm said.