Johnny Russell broke the deadlock at the iPro Stadium in the 67th minute and an own goal from Alex Pearce after 82 minutes ended any doubt about the final outcome.
Steve McClaren’s home side dominated and, despite making six changes to the team that drew with Cardiff City on Saturday, looked fluid and attacked with a good tempo, with on-loan Liverpool winger Jordan Ibe catching the eye on his first start for the club.
The hosts looked lively early on as Will Hughes and Craig Bryson went close with efforts from just outside the area.
Leon Best, from a little further out, at least tested Mikkel Andersen in the visiting goal, as he pushed a drive away. From the resulting corner Best then headed over from eight yards out.
An error by Cyrus Christie just after the half-hour mark let in Jamie Mackie but Kelle Roos denied him with a fine save, low to his left.
Andersen saved from Hughes after 38 minutes, before Ryan Shotton powered a header wide. At the other end, Roos tipped over a Ryan Edwards effort.
Ibe was denied a superb goal six minutes into the second half when Andersen pushed his 20-yard shot on to the bar.
Just as Derby must have been beginning to think it would not be their night, substitute Chris Martin flicked the ball into Russell’s path and the Scot beat two defenders before firing a low shot past Andersen and into the bottom corner of the net.
It was little more than the Rams deserved, but there was more to come.
With their tails up, McClaren’s men continued to pour forward and the impressive Ibe set up Martin whose cross into the six-yard box struck the unfortunate Pearce on the knee and bounced into the net.
The own goal confirmed Derby’s place in the fourth round for the first time since the 2008/09 season, when the Rams reached the last four – where they were defeated over two legs by Manchester United.