A Jordan Rhodes brace helped Rovers secure a first away win of the season at the Goldsands Stadium on Saturday.
Three first half goals put Rovers in total control, but to their credit the Cherries never gave up and grabbed a consolation with 16 minutes remaining.
Rovers took the lead after just 10 minutes courtesy of a Rhodes diving header.
The creator of the first, Leon Best, then added a second as Corry Evans unselfishly squared for the striker to tap home on 14 minutes later.
Rhodes added his second of the game and eighth of the season from the penalty spot after Josh King was fouled by Tommy Elphick to leave the visitors in total control at the break.
Rovers didn't have it all their own way in the second half, despite the home side going down to ten men after Richard Hughes saw red for two bookable offences.
Half-time sub Ryan Fraser sneaking through on goal to beat Jake Kean with 16 minutes to go.
It would have been nice to have followed up last week's first clean sheet with a second, but Rovers will be more than satisfied with their first win on the road.
Rovers made just the one change from the side that drew with Huddersfield Town last time out.
Gary Bowyer welcomed back Leon Best to the starting eleven for the trip to the Goldsands Stadium, Chris Taylor the man making way.
After a quiet first five minutes the game sprang into life with an eventful first half, one that Rovers would dominate.
Jordan Rhodes had threatened on six minutes when he raced through on goal, but he was unable to lift the ball over the advancing 'keeper Ryan Allsop. However on ten minutes Rovers were celebrating with a superb move and a superb goal.
Josh King was a thorn in the Bournemouth side throughout the first half and he set Leon Best away down the left, the striker fired over a clinical cross that Rhodes stooped to head into the far corner via the foot of the post.
Three minutes later and Rovers had huge shouts for a penalty when from a corner Scott Dann's header struck the arm of a defender but referee Rob Lewis was unsighted.
Rhodes tested Allsop with a 16th minute free-kick which he saved well, but the 'keeper had no chance as Rovers added a second just past the midway point of the half.
Tom Cairney found space and time to slip Corry Evans through on goal and with just the 'keeper to beat the Northern Ireland international showed his unselfish side to square for Leon Best to tap home into an empty net.
The Cherries looked to get back into the game, Tokelo Rantie the man the likeliest to do so but Scott Dann, Grant Hanley and Tommy Spurr all defended well when called upon.
But it was at the other end where play was squarely centred, Rhodes and Best both going close.
Four minutes before the break Bournemouth captain Tommy Elphick clumsily brought down speedster Josh King as he raced into the area for a clear penalty. Rhodes making no mistake from the spot as Allsop dived in the opposite direction.
Elphick had escaped a card when he brought down King for the penalty, but he was never going to manage that twice in a row after felling King outside the area moments later, but from Cairney's outswinger Tommy Spurr headed wide.
It was a half in which Rovers had dominated and perhaps should have had more goals to show for it.
Eddie Howe made a change at the break as Ryan Fraser replaced Brett Pitman, but it was Rovers who started on the up once more, Leon Best firing wide of goal on two occasions in successive minutes.
Veteran midfielder Richard Hughes had been introduced as a first half substitute on 32 minutes, but by 51 the former Portsmouth man would be off. Booked for a high boot on Tom Cairney on 46 minutes, he followed that up with a clumsy challenge on Corry Evans which resulted in the inevitable.
Three goals and a man down the home side could have been forgiven if they let their heads drop, but to their credit they continued to try to attack Rovers. Mark Pugh going close with a lovely left-footed shot on the turn that struck the post.
With an eye on keeping their full compliment of players on the park, Gary Bowyer made a double change just past the hour mark.
Leon Best and Corry Evans had both been booked in the first half and the pair, who had combined so well for the second goal, were replaced by Markus Olsson and Lee Williamson on 63 minutes.
Jake Kean had been a spectator for long spells in the game but on 66 minutes he pulled off a superb point-blank save to deny Elphick. As Marc Pugh had fired over a free-kick from the right the Cherries captain met it first time on the volley but Kean beat the ball away.
Rovers responded with another great chance of their own, Williamson found Cairney who played in King round the back, his ball across the six yard box just needed a blue boot to connect and it would have been a fourth, alas it was the red of Bournemouth who cleared. From the corner King turned well but lashed a yard wide.
The visitors looked in control, playing the ball around comfortably but complacency got the better of them on 74 minutes when after a spell of possession it was given away cheaply and Ryan Fraser got to Rantie's pass ahead of Scott Dann to nip it past the Rovers captain and then past the onrushing Jake Kean to make it 3-1.
The Cherries sensed an unlikely way back into the game and didn't give up going forward, but it was Rovers who could have added more as a hat-trick hunting Jordan Rhodes fired wide, whilst Tom Cairney was denied at the near post.
A thousand Rovers fans made the long journey to see their heroes and they were very much rewarded.
AFC BOURNEMOUTH: Allsop, Francis, Pugh (sub Coulibaly 81 mins), Pitman (sub Fraser 46 mins), Grabban, Cook, O'Kane, MacDonald (sub Hughes 32 mins), Ward, Elphick (c), Rantie.
Subs not used: Flahavan (GK), Harte, Surman, Addison.
Goals: Fraser 74 mins.
Booked: MacDonald, Elphick, Hughes.
ROVERS: Kean, Kane, Dann (c), Hanley, Spurr, Cairney, Lowe (sub Marrow 84 mins), Evans (sub Williamson 63 mins), King, Best (sub Olsson 63 mins), Rhodes.
Subs not used: Eastwood (GK), Kilgallon, Taylor, Morris.
Goals: Rhodes 10 mins, 41 mins (pen), Best 24 mins.
Booked: Evans, Best, Lowe.
Attendance: 9441 (997 Rovers fans).