Newly-appointed Cherries boss Quinn saw Jeff Goulding claim a debut goal while former Pirates midfielder Sammy Igoe and Danny Hollands completed a remarkable result just 24 hours after Kevin Bond's sacking.
Youth team coach Joe Roach was given temporary charge following Quinn's appointment at Dean Court and, after his team-mates missed three early chances, Goulding found the way to goal 10 minutes before the break by connecting with Darren Anderton's flick-on to finish from close-range.
Byron Anthony spurned Rovers' best chance when he placed a header onto the roof of the net moments after half-time but 20 minutes from time the Cherries doubled their lead.
Josh McQuoid was sent through by Hollands' incisive pass who pulled the ball back for Igoe to claim his first goal in a Bournemouth shirt.
The home side added a third as Steve Elliott's defensive header landed in no-man's land between defence and goal and Hollands finished convincingly.