Fletcher, a 55th-minute replacement for Donal McDermott, smashed home Bournemouth's winner three minutes from time.
Rovers made the perfect start by scoring in the fourth minute when Gavin Williams set up a shooting chance for Wayne Brown to score.
Williams hit the Bournemouth crossbar five minutes later and then came two red cards, one for each side. Bristol Rovers defender David McCracken was the first to go for elbowing Bournemouth defender Adam Smith and then home midfielder Danny Hollands was dismissed for a second bookable offence.
Bournemouth stepped up the pressure near the end and it paid off in the 84th minute when another substitute, Frenchman Mathieu Baudry, headed in a corner from Liam Feeney.
Fletcher's moment of glory came with time running out after Rovers goalkeeper Conrad Logan lost possession to Danny Ings.
Source: PA
Source: PA