Bradley parred the first extra hole to edge Englishman Rose after the pair had finished regulation at 20-under-par 260 at waterlogged Aronimink.
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.
The win lifted Casey to 12th in the world rankings, although he has since slipped to 16th and has not managed a top-20 finish on the PGA Tour since being runner-up to Bubba Watson at the Travelers Championship in June. Whereas TV coverage was supposed to shift on Sunday from Golf Channel to NBC, for Monday it will remain on Golf Channel after a late start and there will be no PGA Tour Live online streaming at all. He's 36th in the FedEx Cup standings and will miss the Tour Championship.
The pick was to be announced a short time after the BMW Championship ends, whenever it ends.
The Queenslander posted third-place results in the opening two events of the playoffs but couldn't sustain his momentum at the BMW Championship. Tiger Woods shot 66 and was five back.More news: Syria tripartite summit 'step to peace': Iran
"I would love to have got there by winning..."
The top 30 in the standings after this event earned spots in the season-ending Tour Championship, to be played September 20-23 at East Lake Golf Club in Atlanta.
"Yeah it has been a goal of mine all season to make it to the Tour Championships, so I'm pretty proud to pull it off", Smith said.
Unfortunately for the three-time major champion, he finished in a tie for 54th this week. "Was riding some momentum (after the Dell Technologies Championship) but all in all, just game kind of got a little off".
On a course where he once won the AT&T National, Rose shot a 3-under 67 on Monday to get into a playoff with Keegan Bradley.
Bradley went on to win one more time on tour in 2012 and was also named to two Ryder Cup teams in 2012 and 2014 as well as one Presidents Cup team in 2013. "This is one of our tougher tests but it just wasn't playing that way this year, it was just soft and it was hot early in the week, the ball was flying, guys were hitting driver-wedge every hole".