The 34-year-old bogeyed his first hole of the day and had to settle for par on the next 10, but it looked as though he could finally claim his maiden PGA win as he produced a run of three birdies in four holes from the 12th.
“I’m getting close, that’s the good side of it,” said Fleetwood. “On the back nine, I managed to get myself ahead, played some really good golf, putted really well.
“Justin and JJ have done great there and I just didn’t do enough.
“I’m obviously disappointed. I was right in there with a chance, and it’s hard.”
Fleetwood has the highest PGA Tour earnings without a victory, with his runner-up finish to Keegan Bradley at the Travelers Championship in June taking him past $31.4m (£23.3m).
He has now had six top-five finishes this season and 29 on the US tour – 11 more than any other player without a win over the past 40 years, ahead of Brett Quigley with 18.
Rose became the first player in his 40s to win on Tour this season, claiming his first title since the Pebble Beach Pro-Am in February 2023.
He trailed by one shot overnight and had a mixed round until a run of four straight birdies from the 14th.
Rose almost made it five on the last, which would have secured victory without needing a play-off, but his putt was fractionally off target, meaning he finished with a three-under 67.
US Open champion Spaun hit a five-under 65 in the third round and the world number eight repeated the feat to secure his play-off spot after birdies on the 16th and 17th.
“That was an amazing last 90 minutes,” said Rose. “I never stopped believing. I played unbelievable golf coming down the stretch. I had so much fun with it.
“When I bring my best I know I’m good enough to play, to compete, and now to win against the best players in the world.
“It’s a very gratifying day for me and a lot of hard work coming to fruition.
“I feel like there could be a good run of golf [for me] still. I can’t let my age become too much of a story but today is huge for me.”
The top 50 players in the FedEx Cup standings after Sunday’s round now advance to next week’s BMW Championship – the second of the PGA Tour’s three play-off tournaments.