American Bradley, who won his first major crown at last year's PGA Championship, coolly sank a 15-footer to save par on the final hole for a six-under 64 and a 13-under total of 267 at a rain-softened Firestone Country Club.
It was an agonising end for tour veteran Furyk who had led the tournament from the first round and it came after he fell out of a tie for the lead with three holes to play at the U.S. Open in June.
"This just feels so great," an elated Bradley told reporters after landing his third PGA Tour title and the winner's cheque for $1.4 million. "I just felt so strong coming down the stretch. I just enjoyed every minute of it.
"My hope standing on the 18th tee was to make birdie and maybe force a playoff, but you just never know what's going to happen, especially with that pin on 18. That's a really difficult pin."
Bradley all but sewed up an automatic spot on the 2012 U.S. Ryder Cup team with his come-from-behind triumph.
"I just want to be on that team so badly," said the 26-year-old. "I hope this is enough to get on there."
Bradley had been in trouble playing the par-four last, finding a plugged lie in a greenside bunker with a wayward approach and splashing out well beyond the hole with his third shot.
However he coolly sank the par putt coming back before pumping his right fist in delight, then watched in amazement as a distraught Furyk butchered the hole.
Overnight leader Furyk missed a five-footer for bogey to take the tournament into a playoff and tapped in for a six and closing 69 to finish at 12 under with fellow American Steve Stricker, who birdied the last three holes for a 64.
South African Louis Oosthuizen, alone in second place after the third round, was a further stroke back in fourth after carding a four-birdie 69.
Seven-times champion Tiger Woods, who had struggled with his putting in the first two rounds, signed off with a 66 to tie for eighth at four-under 276.
Furyk, seeking a 17th PGA Tour title after firing a best-of-the-week 63 in the opening round, appeared to be heading for a wire-to-wire victory as he played the final hole.
Like Bradley, he miscued his approach and his ball ended up in the rough on the edge of the right greenside bunker from where he had an awkward stance, and chunked his chip to just short of the fringe.
From there, he chipped to five feet short of the hole before missing the putt to hand the title to Bradley.
"I've lost some tournaments in some pretty poor fashions, but I don't think I've let one ever slip nearly as bad as this one," Furyk said. "This was my worst effort to finish off an event."
Bidding for his first PGA Tour title in two years, Furyk led by three shots midway through Sunday's final round before Bradley mounted a spectacular charge with a blistering run of three birdies in five holes after the turn.
Four shots off the pace overnight, Bradley covered the back nine in four-under 31 to end an unforgettable week in the best possible way for his PGA Championship title defence which starts at Kiawah Island in four days' time.
The final round was halted early on by just under three hours due to the threat of lightning before Furyk fended off an early charge by 2010 British Open champion Oosthuizen as both players birdied the first two holes.
Furyk then moved two ahead by sinking a 10-foot birdie putt at the par-four third and Oosthuizen's title bid effectively ended after he bogeyed the fifth and ninth.
Furyk reached the turn in two-under with a commanding three-shot cushion but, at that point, Bradley launched his assault. He drained a 24-footer at the 10th, a seven-footer at the 11th and a 10-footer at the 14th to trim Furyk's lead to just one shot.
Furyk sank a curling 18-footer to birdie the par-five 16th, pumping his right fist in celebration, but remained a stroke in front when Bradley followed suit with a 10-footer.
Two holes later, the tournament ended in dramatic fashion as Furyk imploded and Bradley courageously saved par to triumph by one shot.