McIlroy recorded his first birdie of the day at the tenth but was only able to record one more - at the 16th - to finish three strokes behind Rose and Bradley.
Bradley won his fourth career title, his third in a playoff, to deny Rose his third triumph of the season even though the runner-up effort was enough to see the Englishman atop the rankings for the first time in his career.
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.
Bradley, 32, burst onto the scene with a major victory at the 2011 PGA Championship, but until Monday had not won since 2012.
"I need to start really lifting and getting after it", Woods said of his offseason.
This was a good week for Woods, who was named to the Ryder Cup team and will have some time to rest ahead of the Tour Championship and matches at Le Golf National.
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England's free spirit and attacking flair were again evident in their first match of the UEFA Nations League against Spain at Wembley.
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Darron Lee returned an interception for a touchdown, and Andre Roberts scored on a 78-yard punt return 67 seconds later. All three phases being successful, scoring on all three phases. (Those are) a couple of pretty good defenses.
Rose navigated the rain-soaked Aronimink Golf Course 24 hours after the tournament was almost stopped after 54 holes but fell short of winning the event in a play-off against Keegan Bradley. Rose ended up missing the putt, giving Bradley the win.
"I felt so calm today".
It was incredible there was any play possible at all in Philadelphia, with an inch of rain overnight saturating the course, but officials deemed that play was possible and the first groups took to the course just after 7.30am local time.
Spieth is at 23 and needs to get to East Lake (the Ryder Cup counts as a start) to reach the magic number.
It was the sixth time in Woods' career that he shot 263 or lower over four rounds, and the first time he did not win that tournament. "I was just trying to play".
As Bradley thrust his arms into the air, his family joined him in celebration on the green. Spieth also must qualify for the season finale to play the minimum 25 events this season or face a fine.
By posting a 73 Monday to share 55th, Spieth was one spot outside the qualifiers.
By virtue of his return to East Lake - his first trip since 2013 - Bradley will once again reap the benefits of being a top-30 player on Tour. "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".