But Rose subsequently dropped a shot at the par-four 18th, where his 15-foot putt for the win lipped out.
After Sunday's wet weather postponed the final round, overnight leader Rose got off to a shaky start in defence of his one-shot advantage with a bogey on the first hole.
"I've played a few tournaments (this year) where I was close at the end", Woods said, whose missed opportunities to win this year include the Arnold Palmer Invitational, British Open and PGA Championship.
As consolation prizes go, it was a pretty spectacular one, and just reward for his incredible consistency over the last 18 months.
He says he's speechless and it took a lot of hard work and he doesn't know what to say because he's so happy.
Bradley, 32, burst onto the scene with a major victory at the 2011 PGA Championship, but until Monday had not won since 2012. Rory McIlroy (68) finished two shots behind, while Tiger Woods (65) was three back.
"That's an fantastic achievement, something I'm extraordinarily proud of", he said.
Spieth was keen to play down the severity of the situation after his disappointing final-round 73 in Philadelphia, but he hinted he was expecting to be hit by a fine rather than a ban. I talked to the Tour a little while back and I didn't really think much of it.
His joint sixth-place finish at Aronimink moved him to 20th in the FedExCup Playoffs standings, and comes just days after he was named as a wildcard pick by US Ryder Cup team captain Jim Furyk.
Meanwhile, Justin Rose said he would have like to reach the number one position by winning the BMW Championship. "I'd hate to round off the season with three seconds, but that could be good enough next week, you just never know".
Spieth did not add a new event, and he has only played in 23 tournaments this season. "It's so great", an emotional Bradley said in a greenside interview after securing his fourth PGA Tour victory.