Woods, playing his 11th US PGA event in a comeback from spinal fusion surgery, made four birdies in a row and sank a 25-foot birdie putt at the ninth. But Ancer matched the course record with a bogey-free eight-under par 62, his US PGA career-low score, to finish on 13-under 197, level with Italy's Molinari after 54 holes at TPC Potomac.
A group after Ancer closed with a nine-foot birdie putt, 14-time major winner Woods found a bunker at 18 and missed an eight-foot par putt, taking his fourth bogey to shoot 68 and share 10th on 203.
"It was frustrating because I played better than what my score indicates," Woods said. "I figured I had to get to 10-under and wasn't able to do it. Those four bogeys kind of ruined it."
Tournament host Woods, who hadn't played in the event since 2015 due to back pain, is still contending for the first time since sharing second at Innisbrook and fifth at Bay Hill in March.
"I'm going to have to shoot a low round to try to give myself a chance," said Woods. "I would hope that it would be drier because it gives me a little better chance."
Everyone could face lightning-fast greens on Sunday as sweltering heat is expected to continue.
"I don't have a problem with my strength and my endurance," Woods said. "It's just a matter of keeping focus out here in this heat."
Woods, whose 79 career US PGA victories are three shy of Sam Snead's all-time record, has not won a title since the 2013 WGC Bridgestone Invitational -- a drought of 1,791 days in which 114 different rivals have taken victories.
If Ancer can make it 115, the $1.2 million top prize will eclipse his career $947,637 US PGA prize money.
World number 203 Ancer struggled in his 2016 rookie PGA season and lost his tour spot, but regained it for this season and shared eighth at Houston for his top PGA finish.
"It feels great. I played awesome," Ancer said. "I didn't really make any mistakes, made some putts, kept the ball in front of me. I tried to make as many birdies as I could without making any mistakes."
Molinari closed with back-to-back birdies to shoot 65 and share the lead, two strokes ahead of Americans Ryan Armour and Zac Blair on 199 with Andrew Landry on 200 and Kevin Streelman, who set the course record Friday, sixth on 201.
Woods, 42, is competing at Potomac for the first time. He won the National in 2009 and 2012 at nearby Congressional Country Club.
Woods opened with a bogey, finding deep rough on his first two shots, then rescued par from bunkers at the second and third, praising his new mallet putter.
"It turned everything around making that par at three," Woods said. "I just got rolling after that. I felt good all week with my putter."
Woods began his birdie barrage with seven-footers at the fourth and fifth, sank a 12-footer at six and made an eight-foot putt at seven to reach 8-under, one off the lead.
Another stumble came at the par-4 eighth as Woods found deep left rough and blasted beyond a cart path behind the green on the way to bogey. Woods responded, however, with a dramatic 25-foot birdie putt at the par-3 ninth, giving a double fist pump of satisfaction as the crowd erupted in cheers.
At 13, Woods went beyond a right-side cart path, then over the green before chipping to the far greenside edge and two-putting for bogey. Woods answered by sinking a five-foot birdie putt at 16, but stumbled again at the finish. Ancer birdied four of the first seven holes and made his longest birdie putt of 16 feet at the 16th while Molinari's longest was a 28-footer at the par-3 17th.