Woods, a co-leader after the opening round, battled through a roller-coaster day that featured five birdies, a bogey and a double bogey to maintain a share of the top spot with a two-under 68. Rose posted a three-under 67 to join Tiger in first place.
Rory McIlroy sits in third at five under, while the quartet of Justin Thomas, Jon Rahm, Billy Horschel and Patrick Cantlay are each at four under in fourth place.
Woods came alive on the back nine following a sluggish start. He struggled off the tee early, connecting on just two of his first eight fairways, which forced him to spend more time grinding out pars than seriously challenging for birdies as he did during Thursday's 65.
The 14-time major champion showed a surge in form after the turn, highlighted by three birdies in a four-hole stretch starting at the 12th to open up a two-shot lead.
A wayward drive on the 16th followed by a poor approach shot from the rough eliminated all that progress, however, as he carded a double bogey.
Here's a look at some of his top shots from Round 2:
Not only is Rose in firm contention to win the Tour Championship, but he's also in the driver's seat in the FedEx Cup Playoffs because Bryson DeChambeau is struggling this week. DeChambeau, who entered the event in the points lead, is tied for 27th in the 30-player field at six over.
So that puts Rose in a situation in which there are multiple scenarios that would allow him to claim the postseason title without having to win the year's final tournament. He's the projected champion through 36 holes.
The 2016 Olympic gold medalist tallied five birdies and two bogeys Friday.
Let's check out some of the day's other highlights:
Looking ahead, Tiger had a brief opportunity to put some distance between himself and the rest of the field before his double bogey. So the PGA Tour season finale remains wide open heading into the weekend, which means there's still plenty to decide in the FedEx Cup Playoffs, too.
Coverage of Saturday's third round begins at 12:30 p.m. ET on Golf Channel before switching to NBC at 2:30 p.m. ET.