Rovers were quickly into their stride at the Goldsands Stadium, as a superb header from Jordan Rhodes was followed by a tap-in from Leon Best.
A third followed just before the break when Rhodes converted from the spot after Josh King was felled by Tommy Elphick.
Rovers couldn't hit the same heights in the second period, especially after Richard Hughes saw red after too bookings in the space of five minutes.
The home side got their reward in the shape of a first goal for Ryan Fraser, the substitute nipping in to score a consolation in the final quarter.
"I thought we were excellent in the first half," said the manager after the game.
"We spoke in the week about getting the first goal and also getting the first away win and the game plan, in the first half, all the lads carried it out superbly and I thought we were very good in our pressing game and then we were very good in our attacking play.
"They are a young team that are learning and are starting to learn very quickly now, we are delighted with coming here because it is a difficult place to come."
Rovers couldn't add to their three goals in the second period, with the manager upset that they couldn't claim back-to-back clean sheets.
"I think you have to give credit to Bournemouth, they changed their system. But we are disappointed that we didn't defend their goal better and keep a clean sheet.
"At the start of the game the outcome was to try and get three points and get our first away win and we are happy that we achieved that today."
Watford are Rovers' next opponents on Tuesday night at Ewood Park.
Tickets for the 7.45pm kick-off are on sale right now, priced from £15 for Adults, £10 Seniors and £5 Juniors.