Both Bradley and Rose came to the 18th hole at 21-under before Bradley, playing in the group ahead of Rose, made bogey to open the door for Rose.
He is now projected to finish 31st in the FedExCup standings and miss out on the Tour Championship as only the Top 30 players advance to the season finale at Atlanta's East Lake.
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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.
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. I turned pro at 18 - it took me 20-odd years to get to No. 1. Several of the top golfers in the world are competing for the prize money and to put themselves in position for the final event of the FedEx Cup playoffs, the Tour Championship. His first putt bounced in the fringe and finished five feet short, then his par effort was left the whole way.
After several attempts to get play underway on day four of the BMW Championship, play was abandoned for the day with organisers of the third FedEx Cup Playoffs event still hopeful of a Monday local time finish. "When you go in there 15th, 16th or 17th, you know you have to win, and then you have to hope for a ton of other things to happen".More news: Tropical Storm Florence continues its churn toward USA coast
"I assume it will either be a fine or I'm adding some in the fall, " Spieth said.
And this was the kind of day when more than one mistake was going to be costly.
But not getting a final round would obviously affect several players who are trying to finish among the top 30 in points and advance to the season-ending Tour Championship that begins September 20. Woods finished the regular season 20th in FedEx Cup points and had slipped to 25th heading into the BMW.More news: Is Gareth Southgate ready to leave England job after Euro 2020?
He will have an unexpected week off before joining the American team for the September 28-30 Ryder Cup in Paris.