Warren Cummings' right-footed cross found Pearce who managed to bundle home his 75th minute effort to win a bad-tempered game.
Rovers defender Byron Anthony saw red for two yellow cards in the first-half, while Bournemouth's veteran striker Lee Bradbury got his marching orders in the 78th minute for a second bookable offence.
There were seven other bookings in the match as referee Clive Penton struggled to keep control.
Brett Pitman almost opened the scoring in the third minute, but his header was kept out by Steve Phillips in the Rovers goal.
Phillips then kept out a Darren Anderton free-kick just before the break to keep his side in the game.