It was set up to be a close contest against fellow championship opposition, but after Colin Kazim-Richards' opener on the stroke of half-time there was only ever going to be one winner at Pride Park.
The second half was virtually one way traffic towards the Derby penalty area, a thunderous strike from Scott Dann and a delicate finish from top scorer Jordan Rhodes ensuring a pleasing day for Appleton.
"I think from a performance point of view and especially an away performance it is up there as almost being perfect," said the Rovers boss after the game.
"The first 20-25 minutes I thought it was difficult for no other reason than the exploits the players showed on Tuesday night, but once we got through that period we never really looked in any threat.
"I don't think our goal was in any danger at any point and we grew stronger in the game and some of our experienced players came to the fore."
As with the midweek match against Brighton, Rovers scored on the stroke of half time, Colin Kazim-Richards out-muscling Derby captain Richard Keogh to race through to score in fine style.
"It is a good habit to have," added the Rovers boss. "We did it on Tuesday which gave us a lift and obviously we did it today. It was a fantastic goal, a great finish from Kaz. It gave us a platform for us to use in the second half.
"Then it was important that we got the second goal, we couldn't get it on Tuesday but we have done today, I thought the game was comfortable after that, Jake hasn't had a lot to do today which was pleasing.
"I'd love a home tie in the next round, it would be great to see a full house at Ewood Park against one of the big Premier League boys to remind people what a fantastic club this is and what fantastic supporters we've got."
The draw for the fifth round of the competition takes place this Sunday evening on ITV after the Oldham v Liverpool match. Rovers will be ball number 4.