Woods, who at one point in Sunday's third round stood 14 strokes off the pace, rolled in a six-foot birdie putt at the seventh hole to reach level-par for the tournament, seven shots behind co-leaders Lucas Glover and Ricky Barnes.
The world number one was one-under for the round when failing light forced an end to play. Players return to Bethpage Black to resume the final round at 0900 (1300 GMT) on Monday.
"I have hit the ball well enough to do it, but just haven't made the putts," Woods told reporters after a third-round 68.
"Today I was within 15 feet a lot, but just did not have the putts to go in. I've been lipping it out, burning the edges and just haven't got it right yet."
Holder Woods, who also claimed the Open crown when it was played at Bethpage in 2002, said he had made only one good putt all day with his birdie at the par-three third.
He birdied the par-three 17th without using his putter -- holing out a chip shot from some heavy greenside rough.
Woods started the day 11 shots off the pace and fell back by 14 before bouncing back to trail by nine at end of the third round. He sliced two more strokes off the deficit after starting the final round.
Woods, winner of 14 major championships, has never won a major from behind going into the last 18 holes.
"It's one of those where you have to play a great round of golf and get some help," he said about winning again at Bethpage.
"Obviously it's not totally in my control."