Christopher Samba and Jason Roberts both found the back of the net inside the opening ten minutes, before Douglas Rinaldi pulled a goal back for Aidy Boothroyd's side. But when Benni McCarthy gave Rovers a 3-1 half-time lead with his 20th of the campaign, the contest was effectively ended.
Mark Hughes named an unchanged side following Sunday's extra-time defeat to Chelsea, whilst Boothroyd made two changes to the side that were humbled 4-1 by Manchester United - with both starting players experiencing differing fortunes early on.
Indeed England international goalkeeper Ben Foster, who was ineligible to play against his parent club at the weekend, was arguably at fault for Rovers' opening two goals.
On seven minutes, David Bentley's corner from the left, which he had won himself, deceived Foster mid-flight and Samba rose to head home his second Rovers goal from close range.
Three minutes later and Rovers doubled their lead. Bentley was again the provider, playing the ball over the top to Roberts, who held off the challenge of Jordan Stewart before stroking the ball across Foster and into the far corner of the goal for his second in as many games.
Morten Gamst Pedersen swerved a left-foot strike well wide of the mark shortly after, before tapping home Roberts' parried shot just a minute later, but the goal was rightly ruled out for offside.
And just as Rovers had scored with their first attack of the game, so did the visitors - on 21 minutes. Hameur Bouazza's deep cross found the recalled Rinaldi and the Brazilian forward headed the ball past Brad Friedel with the aid of a slight deflection off Brett Emerton.
On the half-hour mark, Rovers' appeals for a penalty were waved away after Clarke Carlisle had connected with Roberts inside the box, but the home side added their third of the game just two minutes later.
Bentley claimed his third assist of the half, swinging in a cross from the right and McCarthy got in front of his marker to glance a diving header into Foster's bottom right-hand corner.
Steve Kabba replaced goalscorer Rinaldi at the break and almost reduced the arrears with his first touch, but his fierce drive from 25 yards out was sensationally saved by Friedel.
At the other end, Rovers almost extended their lead when Roberts broke magnificently from his own half to the edge of the opposition box, before playing in McCarthy, but the South African sliced his shot with a clear sight of goal.
Bentley twice fired narrowly wide from long range late on, but the points were already safe as Rovers climbed back into the top half of the table.