The Ryder Cup star had one hand on the title when he birdied 16 and 17 to keep pace with Bradley, and Rose needed only a par at the last for victory when the American pushed his second into the grandstand and failed to get up-and-down.
Keegan's father, Mark Bradley, has been with Jackson Hole Golf and Tennis for years.
He was tied with Xander Schauffele on the 18th hole of the third round Saturday when Schauffele's 8-foot par putt rimmed around the high side of the cup, giving him a bogey and Rose the lead.
Both Bradley and Rose missed the green to the right on the playoff hole.
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Woods bagged four birdies on the opening nine and pulled within one shot of the lead on the back side. He grinned as he tossed his son, Logan, into the air. To win and to put everything together when I didn't know if I was going to win again, to get to the Tour Championship, to have them here and have what's going on in my life is just so wonderful.
Even though Woods prefers tougher conditions this is the second time over the last month he's almost gotten it done in a birdie-fest, including the PGA Championship at Bellerive.
Still, Rose will overtake Dustin Johnson to become the world's top-ranked golfer for the first time, marking the culmination of 18 months of remarkably consistent play by the Englishman.
"Lots of hard work", he said when asked what has gone into this comeback. It was glue coming out of that fringe there and then I under-read the putt a little bit. As the leaders played their final few holes Monday at Aronimink, he gave one of his most telling answers yet about his prospects for next season. Bradley hasn't made the Masters since his exemption for winning the PGA ran out, but he'll be back at Augusta next April. It's tough to go from being on Ryder Cup teams, being on Presidents Cup teams, to outside the top 100 in the world. He's been close to breaking through a handful of times over the last two years, but this one meant something more. Spieth started 27th in the FedEx Cup standings and took a step backward, finishing with a 73 to tie for 55th.
The top 30 on a season-long points list advance to the Tour Championship in Atlanta starting on Thursday week.