Woods was two under par after six holes in perfect scoring conditions, but eventually had to settle for a one-over 71 after running up a double-bogey six on the ninth, his final hole.
That left the world number one, whose last major win came in the 2008 US Open, six shots behind clubhouse leader Jim Furyk, who bogeyed the ninth to card a 65 and miss out on equalling the course record by a shot.
"I'm still right there," said Woods, who won his eighth Bridgestone Invitational by seven shots on Sunday. "As of right now I'm only six back and we have got a long way to go.
"I played really well today. The round realistically could have been under par easily."
Woods certainly looked like being three under when he hit his approach to the second to three feet, but his birdie attempt never even touched the hole.
"(It was a) tricky little putt," said Woods, whose group were being timed for slow play at that stage. "Probably should have taken a little bit longer but we were on the clock and had to get going, but I ended up blocking the putt anyways down the hill.
"We were put on the clock on the first hole so we had to get going and try to close up the gap. It would have been nice to take a little bit longer on that putt.
"Looking from behind the ball it looked like it was going to snap (turn sharply). Looking from behind the hole, it looked like it wasn't going to snap. I ended up playing it somewhere in between and then blocked it.
"We have to be conscious of it (being timed), but we play just about every week like that. We have a lot of people following us and a lot of cameras going off and movement inside the galleries. We have a tendency of having that happen and we're accustomed to it. It happens."
Woods made his first bogey of the day on the par-five fourth after failing to get up and down from a greenside bunker, and then fluffed a pitch on the ninth into more sand to take another six.
"I was just trying to play 20 feet long and putt back and get my bogey and get out of there," he said of the ninth. "If I happen to make a 20-footer, so be it. But it came out nowhere and I didn't get over the bunker."