The world number 15 lost his lead early in the third round following a double bogey at the par-five third and a bogey at the seventh.
That left Rose alone in the lead and he followed with a birdie at the seventh, but then found water at the ninth and made bogey while Fleetwood birdied the ninth to regain a share of the lead.
The 34-year-old went clear again with a 14-foot birdie putt at the 12th and claimed another birdie on the 16th.
He had a three-shot lead after 17 holes, but a bogey on the last allowed Rose to pull within one with a birdie.
Fleetwood has had five top-five finishes this season and was runner-up behind Keegan Bradley at the Travelers Championship in June.
Players who finish in the top 50 of the FedEx Cup standings after Sunday’s round at TPC Southwind will advance to next week’s BMW Championship – the second of the PGA Tour’s three play-off tournaments.