Last season's quarter-finalists were undone by a Warren Cummings cross and defender Pearce smashed the ball into the net.
Anthony saw yellow for diving in the 37th minute and compounded his error when he saw red two minutes later for bringing down Sammy Igoe 30 yards from goal.
Bradbury was booked in the 68th minute and followed Anthony down the tunnel ten minutes later as the game threatened to boil over.
Brett Pitman should have opened the scoring in the third minute but his weak header was easily saved by Steve Phillips.
Rickie Lambert has 15 league goals already notched up this season, but missed his only chance here as Shawn Jalal made an easy save from his header.
Cummings was booked for a clumsy challenge in the 20th minute before Anthony saw red for his two bookings.
Darren Anderton almost added a second before the break for Bournemouth with a rasping free-kick from 30 yards.
Rovers goalkeeper Steve Phillips did well too take the sting out of the shot as Bournemouth threatened to run riot.
The Cherries broke through in the 76th minute through Pearce as the game came to life.
After the Bradbury sending-off Rovers threw every man forward and it was Phillips who created a great opening.
The giant goalkeeper went up for a corner and created havoc in the defence and the ball fell to Craig Disley seven yards from goal.
The midfielder did not strike the ball well and hit the sidenetting as last season's FA Cup quarter-finalists went out in the first round.