Bradley parred the first extra hole to edge Rose after the pair had finished regulation on 20-under-par 260 at waterlogged Aronimink in suburban Philadelphia.
Rose missed a four-foot putt that would have extended the playoff but still became the 22nd different player to hold the world number one ranking.
"I can't believe this. It's so great," an emotional Bradley said in a greenside interview after securing his fourth PGA Tour victory.
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"It took a lot of hard work, a lot of people helping me."
Bradley, 32, burst onto the scene with a major victory at the 2011 PGA Championship, but until Monday had not won since 2012.
His victory capped a long week that stretched to Monday after persistent rain left the course unplayable for the scheduled final round on Sunday.
Bradley carded a closing six-under-par 64 and hardly put a foot wrong until he bogeyed the final hole, a mistake that looked like costing him victory.
But Rose subsequently also dropped a shot at the par-four 18th, where his 15-foot putt for the win lipped out.
The pair finished one shot ahead of Americans Xander Schauffele and Billy Horschel.
Northern Irishman Rory McIlroy finished two shots behind, while Tiger Woods was three back.
Rose got a lucky break in the playoff when his approach shot landed in a skybox but ricocheted back towards the hole and stopped just off the green.
However he misjudged his putt through the soggy grass and it came up well short of the hole, before he missed the par putt.
That left Bradley to tap in for the win.
"I hit a great putt in regulation to win it," Rose said.
"Just a poor playoff. It was glue coming out of that fringe there and then I under-read the putt a little bit. It's over before you know it."
The BMW was the third of four FedExCup playoff events on the PGA Tour.
The top 30 on a season-long points list advance to the Tour Championship in Atlanta starting on Thursday week.
Jordan Spieth was the most prominent player to miss out. He started the BMW ranked 27th but fell to 31st with a poor finish on Monday.
"I was in control of my own destiny but just didn't have it this week," said Spieth after coming tied 55th.
He will have an unexpected week off before joining the American team for the Sept. 28-30 Ryder Cup in Paris.
England's Paul Casey, a member of the European team, pulled out of the BMW after the third round, citing back pain.
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