Teenager Mason Bennett scored the first goal of his career as a rampant Derby eased into the FA Cup fourth round.
The 16-year-old striker came off the bench to fire in the Championship side's fifth goal of a fine display.
Ben Davies poked in to give Derby the lead just before half-time before goals from former Wigan striker Conor Sammon and John Brayford made it 3-0.
Jeff Hendrick prodded in a fourth before Bennett rifled in from 18 yards to complete an emphatic victory.
It was a chastening afternoon for Tranmere, who were making their first appearance in the third round for three years. They started brightly, almost taking a third-minute lead only for centre-back Ash Taylor to head Andy Robinson's floating corner wide.
Rovers continued to create chances, with Derby goalkeeper Adam Legzdins having to hold on to a Donervon Daniels header from Adam McGurk's free-kick.
But visiting keeper Owain fon Williams put his side in trouble, three minutes before half-time when Paul Coutts's shot squirmed under his body and into the path of Davies, who was on hand to tap the ball into the empty net.
The Rams added a second nine minutes after the interval when Sammon blasted into the bottom corner from 12 yards. And they sealed matters on the hour when Rovers failed to clear their lines and Brayford tapped in from close range.
Hendrick, making his first FA Cup start, added a fourth after some neat build-up play before Bennett illuminated the final minutes with a fierce shot that Fon Williams was powerless to stop.
Derby County manager Nigel Clough:
"Mason's goal was the highlight of the day for me. He could have scored one when he made his debut at Middlesbrough and hit the bar so it was lovely for him to get one.
"It's good because I think the experienced lads on the bench all understand that although all of them could have come on, it's vital for the younger ones to come on and get experience.
"One of the challenges we had was to bring some of the younger players through for the first team primarily and if they make us a profit at some point, all well and good, but I think this is the best place for them at the moment.
"To be fair, I didn't think it was a 5-0. We took our chances and got goals we don't normally score when the ball is bobbing around in the six yard box which was pleasing. The first goal was important because it settled everybody down and we went in at half time with a big boost."
Tranmere Rovers manager Ronnie Moore:
"The turning point for me was the first goal. I thought we probably edged the first half so to concede was disappointing for us. We were flat at half time but to be fair, the keeper has been fantastic for us this season.
"He's made some unbelievable saves and he feels deep down he should have saved it but I'm not going to criticise him. He'll come back and hopefully keep a clean sheet next week.
"The basic defending just after half time was poor and really, we gifted them three goals but I think we can take a lot of positives out of the first half.
"I didn't think 5-0 was a reflection on how the game was. The last time we got beat 5-0 we won three league games on the trot."
Live Text Commentary
Full Time The referee ends the match.86:35
The assist for the goal came from Conor Doyle.86:35 Goal scored
Goal! - Mason Bennett - Derby 5 - 0 Tranmere A goal is scored by Mason Bennett from the edge of the penalty box to the bottom left corner of the goal. Derby 5-0 Tranmere.85:51
Free kick awarded for a foul by Danny Holmes on Michael Jacobs. Mark O'Brien restarts play with the free kick.83:32 Substitution
Substitution Ben Davies leaves the field to be replaced by Mason Bennett.83:32
Effort from long range by Adam McGurk goes wide left of the goal.82:04
Foul by Mark O'Brien on Max Power, free kick awarded. Direct free kick taken by Adam McGurk.78:15 Substitution
Substitution Paul Coutts goes off and Conor Doyle comes on.76:03
Jeff Hendrick challenges Cole Stockton unfairly and gives away a free kick. Adam McGurk restarts play with the free kick.74:33
Corner taken by Adam McGurk, Effort on goal by Zoumana Bakayogo from just inside the penalty area goes harmlessly over the target.73:16 Substitution
Substitution (Derby) makes a substitution, with Kieron Freeman coming on for Gareth Roberts.73:16 Substitution
Substitution (Tranmere) makes a substitution, with Paul Black coming on for Andy Robinson.71:32
The assist for the goal came from Ben Davies.71:32 Goal scored
Goal! - Jeff Hendrick - Derby 4 - 0 Tranmere Jeff Hendrick gets on the score sheet with a goal from just inside the area to the bottom left corner of the goal. Derby 4-0 Tranmere.69:30
Unfair challenge on Andy Robinson by Ben Davies results in a free kick. Free kick crossed by Andy Robinson.68:41
Unfair challenge on Max Power by Will Hughes results in a free kick. Free kick taken by Owain Fon Williams.66:31
The referee blows for offside. Owain Fon Williams restarts play with the free kick.65:56
Jeff Hendrick takes a shot. Save made by Owain Fon Williams.64:18
Adam McGurk concedes a free kick for a foul on Michael Jacobs. Paul Coutts restarts play with the free kick.62:35
Assist on the goal came from Ben Davies.
Goal! - John Brayford - Derby 3 - 0 Tranmere John Brayford scores a goal from deep inside the penalty box to the bottom right corner of the goal. Derby 3-0 Tranmere.61:17
Michael Jacobs takes a shot. Save by Owain Fon Williams. Corner taken by Ben Davies,58:02
Cole Stockton has an effort at goal from outside the box which goes wide left of the goal.54:49
Cole Stockton has an effort at goal from just inside the box that misses to the right of the target.53:17
The assist for the goal came from Ben Davies.53:17 Goal scored
Goal! - Conor Sammon - Derby 2 - 0 Tranmere Conor Sammon scores a goal from deep inside the penalty box to the bottom left corner of the goal. Derby 2-0 Tranmere.52:29
Unfair challenge on Paul Coutts by Andy Robinson results in a free kick. Ben Davies restarts play with the free kick.45:01
The second half begins.45:00 +1:28 Half time
Half Time It is the end of the first-half.45:00 +0:44
Liam Palmer is ruled offside. Free kick taken by Adam Legzdins.44:26
Jeff Hendrick has an effort at goal from outside the penalty box which goes wide left of the goal.43:32
Michael Jacobs challenges Danny Holmes unfairly and gives away a free kick. Direct free kick taken by Owain Fon Williams.41:59
Assist by Paul Coutts.41:59 Goal scored
Goal! - Ben Davies - Derby 1 - 0 Tranmere Ben Davies scores a goal from inside the six-yard box to the bottom left corner of the goal. Derby 1-0 Tranmere.35:27
Michael Jacobs produces a cross, Ash Taylor gets a block in. Inswinging corner taken from the left by-line by Ben Davies, Close range headed effort by Richard Keogh misses to the right of the goal.35:09
Michael O'Halloran is penalised for handball and concedes a free kick. Direct free kick taken by John Brayford.33:06
Free kick awarded for an unfair challenge on Paul Coutts by Max Power. Ben Davies restarts play with the free kick.32:54
Effort from 20 yards by Jeff Hendrick. Blocked by Liam Palmer.32:23
Michael Jacobs takes a shot. Blocked by Ash Taylor.28:20
Adam McGurk fouled by Ben Davies, the ref awards a free kick. Free kick taken by Danny Holmes.26:35
Paul Coutts takes the free kick. Michael Jacobs takes a shot. Liam Palmer gets a block in.25:21
Outswinging corner taken right-footed by Ben Davies, Headed effort from deep inside the penalty area by John Brayford misses to the left of the goal.25:03
Free kick awarded for a foul by Donervorn Daniels on Conor Sammon. Paul Coutts takes the direct free kick.23:33
Cole Stockton takes a shot. Save by Adam Legzdins.22:37
Free kick awarded for a foul by Mark O'Brien on Adam McGurk. Adam McGurk restarts play with the free kick.22:08
Shot from a long way out by Paul Coutts goes wide of the left-hand upright.20:50
Max Power gives away a free kick for an unfair challenge on Ben Davies. Ben Davies restarts play with the free kick.18:26
Jeff Hendrick produces a cross, clearance by Ash Taylor.14:18
The referee blows for offside against Conor Sammon. Free kick taken by Owain Fon Williams.12:58
Short corner taken by Ben Davies.11:00
Max Power challenges Ben Davies unfairly and gives away a free kick. Free kick taken by Ben Davies.9:53
Shot by Max Power. Adam Legzdins makes a save.9:03
Outswinging corner taken from the right by-line by Ben Davies, clearance by Max Power.5:22
Ben Davies takes a shot. Save by Owain Fon Williams.2:36
Adam McGurk takes the inswinging corner, Headed effort from deep inside the penalty area by Max Power misses to the left of the goal.1:47
Ben Davies gives away a free kick for an unfair challenge on Liam Palmer. Free kick taken by Donervorn Daniels.0:00
The referee gets the match started.
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