Scott came within a shot of the course record with an 8-under 62, six shots better than the score posted by Woods in his first competitive round in almost three months.
Woods, having recovered from knee and Achilles tendon problems in his left leg, dismissed longtime caddie Steve Williams after Williams worked two tournaments for Scott.
Playing two groups behind Woods on the Firestone Country Club's South Course, and with Williams on his bag, Scott birdied four of the last eight holes and just missed the record of 62 shot in 1990 by Jose Maria Olazabal. Woods equaled that record in 2000 during one of his seven victories in this tournament.
Alone in second was Jason Day at 63, followed by Nick Watney with a 65. Those tied for fourth at 66 in the elite 76-player event included Rory Sabbatini, D.A. Points, Thomas Bjorn, Brandt Snedeker, Pablo Larrazabal, Martin Laird, Kyung-tae Kim, Ryan Moore and former champion Stewart Cink.
Phil Mickelson and Lee Westwood were among those at 67 and Woods was in a large group at 68 that included world No. 1 Luke Donald, Rickie Fowler and U.S. Open champion Rory McIlroy.
Woods, who has fallen to No. 28 in the world rankings, played a solid round that included three lengthy par-saving putts on the front nine.
"I was hitting proper shots out there," Woods said, "I hadn't hit the ball like this (in a while). This was fun, to be able to hit the ball with that much feeling through the golf ball. It was pretty nice."
In what has been typical for Day this season, he came close to being the biggest story of the opening round. In a six-tournament stretch earlier this year he finished second at both the Masters and U.S. Open and was sixth at The Players Championship. That run boosted him to No. 8 in the world.