Summer signings Michel Salgado and Niko Kalinic both scored their first goals for the club as Sam Allardyce's side comfortably overpowered Peterborough United's ten men in the end.
For 40 minutes, Darren Ferguson's Championship strugglers gave a decent account of themselves.
However, the dismissal of keeper Joe Lewis, who was red-carded for a foul on Morten Gamst Pedersen shortly before half-time, left the visitors with a mountain to climb.
Rovers then ruthlessly made the most of the extra man, running in three second-half goals to ease into the last eight.
The prospects of a giant-killing looked remote when Rovers surged ahead after just four minutes.
A foul on Junior Hoilett led to a free-kick on the edge of Peterborough 's penalty area and keeper Lewis was promptly left rooted to the spot as Pedersen curled an exquisite shot into the far top corner.
That was a hammer blow for the Posh but, to their great credit, they dug deep and hit back with an equaliser in the 17th minute.
Tommy Rowe bypassed Rovers' central defenders with a raking cross to the far post and Chris Whelpdale ghosted in undetected to guide a header carefully past Jason Brown.
That goal filled the visitors with confidence and for a while they looked threatening going forwards.
However, all of that good work was then undone when Russell Martin made a costly error three minutes before the break.
The defender's attempted backpass fell woefully short, leaving Pedersen in a one-on-one with Lewis and the Norwegian was promptly brought down in the area.
Referee Anthony Taylor gave Lewis a straight red and awarded a penalty, which was dispatched with aplomb by Steven Reid.
Despite that double setback, United conjured a second equaliser in the 50th minute, George Boyd thumping a fierce left-footer past Brown from just inside the area, following good work by Whelpdale.
However, the scores were level for just six minutes before Salgado restored Rovers' advantage, the Spaniard expertly taking a pass from Brett Emerton in his stride and then beating substitute keeper James McKeown at his near post.
After that, it was all one-way traffic as Rovers made the most of their numerical advantage.
Benni McCarthy added a fourth in the 72nd minute.
Then Kalinic bagged his first goal since a summer move from Hajduk Split three minutes later, after Rovers were awarded another penalty for a handling offence.