A Jeff Hughes penalty, following a 33rd-minute foul by Adam Smith on Gavin Williams, settled the outcome.
Rovers also hit the crossbar through a second-half Chris Lines free-kick before having to withstand a late onslaught from the Cherries.
A delighted Campbell said: "The effort, determination and desire we need was there in abundance. It was clear for everyone to see how much the game meant to us.
"You couldn't single any player out. All of them put their bodies on the line and it was a fantastic backs-to-the-wall performance.
"We've been in the bottom four for far too long but while we will celebrate, the result will be put behind us [on Wednesday] and everyone will be concentrating on Saturday's game with Exeter City."
Source: PA
Source: PA