Woods, making his first appearance in the PGA Tour’s season finale since 2013, followed up his opening-round 65 with a two-under 68 to move alongside Rose on seven under.
The pair hold a two-shot cushion over nearest challenger Rory McIlroy, who mixed five birdies with three birdies to match Woods’ 68, with defending FedExCup Justin Thomas one of four players sitting three off the pace.
Billy Horschel, Patrick Cantlay and Jon Rahm sit on four under alongside Thomas, with overnight joint-leader Rickie Fowler a further stroke back after a two-over 72.
Fowler – who began the day alongside Woods – recovered from an opening-hole bogey to drain a 25-footer at the next and post back-to-back birdies from the fourth to reach the turn one ahead.
One-time leader Rahm birdied his first three holes on his way to a front-nine 30, while Woods holed from 20 feet at the second but missed from six feet to save par at the ninth.
Woods made a five-foot birdie at the 12th – where Fowler made his second bogey in three holes – to move into the outright lead, only for Rose to follow a gain at the 13th with a 30-footer at the next to get to seven under.
The 14-time major champion responded by converting successive long-range gains from the 14th, which briefly opened up a two-shot lead when Rose failed to convert an eight-footer for par at the 16th.
Woods double-bogeyed the 16th after finding rough of the tee and then plugged in a bunker with his second shot, as Rose got up and down from sand to save par at the 17th and made a tap-in birdie at the last to set the clubhouse target.
Needing a final-hole birdie to join Rose, Woods attacked the par-five green in two and posted a two-putt gain from 30 feet to get to seven under and ensure he will be out in the final group on Saturday.
Watch the Tour Championship throughout the weekend live on Sky Sports. Live coverage continues on Saturday from 5.30pm on Sky Sports Golf and 8pm on Sky Sports Main Event.