Rose held a one stroke advantage going into the final hole but a bogey meant it went down to a playoff.
Currently, Woods leads the PGA Tour with a 68.42 scoring average in the third round, and average that ranges from 46th to 133rd in the other three rounds.
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.
But guaranteeing a top 30 position in the final FedEx Cup points rankings, Bradley secures his spot in the field at the major championships in 2019, something the former college golfer at St. John's has struggled with ever since his exemption for winning the 2011 PGA Championship expired.
Of course, his principal aim on Monday in Pennsylvania was to win the rain-delayed BMW Championship at Aronimink and as his putt from 12 feet traversed its sweet course to the cup on the 18th in regulation play, there was seemingly nothing stopping him.
The pair finished one shot ahead of Xander Schauffele (67) and Billy Horschel (64).
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Webb Simpson, Jason Day, Open Championship victor Francesco Molinari, Phil Mickelson, Masters champion Patrick Reed, Patrick Cantlay, Rory McIlroy, Xander Schauffele, Tommy Fleetwood, and Tiger Woods.
"To get to world No 1 is unbelievable, and I can say now in my career I've been the best player in the world", said Rose afterwards.
Mathematically, Woods still has a chance to win the FedEx Cup, although it would require several top players to lose form. It was glue coming out of that fringe there and then I under-read the putt a little bit.
"The name of the game is you've got to make putts and you've got to roll it", Woods said after the second round, according to ESPN. He started the BMW ranked 27th but fell to 31st with a poor finish on Monday.
Players are required to play in 25 events in a season or risk a fine; Spieth played in 24 events and was clearly banking on making East Lake his 25th.
"I assume it will either be a fine or I'm adding some in the fall, " Spieth said.