James Henry, who arrived in the week from Reading, scored in each half.
The first was a stooping header just before half-time and the second a thumping drive in added-on time.
Neither of these clubs has yet won at home and Rovers' record of now ten games without a win means they have been sucked into the relegation zone.
Their lack of confidence was on early display when Sean Rigg, promoted to the starting line-up for his pace, went by Bournemouth's centre-back Josh Gowling and then made the wrong decision. He tried and failed to lay off a pass when a shot was the right option.
His time on the pitch was short as a centre circle collision after 20 minutes saw him go off with double vision.
After weathering early pressure and a couple of speculative efforts from Lewis Haldane, Bournemouth gradually took over.
They had an anxious moment when David Pipe on the right flank dropped an intelligent ball between centre-backs Gowling and another Reading arrival Alex Pearce.
Goalkeeper Neil Moss sprinted outside his area to boot the ball away from Rigg, while Pearce quickly tightened up his game and, despite a booking, had a solid debut.
Meanwhile, Henry, was wide on the right and enjoying himself. Once fierce drive bounced away off the keeper's knees and then he timed his run perfectly for his first goal to head past Steve Philliips as Jo Kuffour crossed from the left.
But this match belonged to Darren Anderton. The one-time England player strolled through the midfield with acres of room because no-one picked it up.
It was his ball which set up the first goal, but Henry's second saw him push the ball between the tiring legs of Rovers' new loan signing Danny Coles.
He then skipped round the Hull City player, who has joined Rovers on a two-month deal, and thumped his shot high into the net.