The 16-year-old striker, an 84th-minute substitute, made an immediate impact as he fired in Derby's fifth to round off an emphatic display from the Championship outfit.
League One leaders Rovers, 15 places below the Rams, initially started well but once Ben Davies capitalised on goalkeeper Owain Fon Williams' mistake in the 42nd minute there was no way back.
Former Wigan striker Conor Sammon scored a second - his seventh of the season and first in the competition - and John Brayford made it a trio of goals in 21 minutes. Jeff Hendrick added a fourth before Bennett struck to round off the scoring.
Rovers created some chances early on, with Adam Legzdins having to hold on to a Donervorn Daniels header from Adam McGurk's free-kick.
Fon Williams put his side in trouble, though, by picking up a backpass in the 28th minute, although the shot-stopper was relieved to see his team-mates block Michael Jacobs' effort following the free-kick.
However, Fon Williams' luck ran out three minutes before half-time when Paul Coutts' shot squirmed under his body and into the path of Davies, who was on hand to put the ball into the empty net.
County added a second nine minutes after the interval when Sammon stole the ball away from a defender and blasted into the bottom corner from 12 yards.
Rovers still had some fight left in them and almost halved the deficit a minute later, but Cole Stockton cut in from the left and fired narrowly over.
But, after Fon Williams tipped Jacobs' long-range effort wide, Derby put the game to bed on the hour mark when Rovers failed to clear their lines and Brayford tapped in from close range.
The Tranmere goalkeeper had to be alert to keep out Hendrick's 20-yard strike, only for the midfielder - making his first FA Cup start - to find the net five minutes later after some neat build-up play.
And Derby scored a fifth four minutes from time when Bennett, who replaced Davies, rifled in from 18 yards.