Derby 0-2 Stoke
Goals from Cameron Jerome and Robert Huth ensured last season's FA Cup finalists Stoke City safely negotiated their way into the fifth round.
Striker Jerome's close-range effort in the fifth minute gave the visitors the lead after a nervy start from Derby.
The Championship side did enjoy periods of possession but struggled to test Stoke goalkeeper Asmir Begovic.
And in the 81st minute, Huth volley through a posse of players to deceive Begovic's counterpart Frank Fielding.
Derby had won their previous five home league and cup matches, but that form deserted them against a side eighth in the top-flight and preparing to play Valencia next month in the Europa League.
Nigel Clough's side went behind when they were undone by a short corner involving Matthew Etherington and Glenn Whelan.
Whelan delivered a low ball into the heart of the area, which was touched on by Marc Wilson before hitting Huth.
The German defender appeared poised to smash the ball into the net, but instead it was Jerome who prodded home for his third goal in as many appearances, and his fourth in the last five.
Derby managed to create just one chance in the first half when a Dean Whitehead header fell into the path of Jeff Hendrick, who cracked an 18-yard half-volley narrowly wide.
The introduction of substitutes Jermaine Pennant and Kenwyne Jones midway through the second half provided Stoke with greater impetus.
From a 35-yard free-kick in the 73rd minute, Pennant brought a stunning save from Derby keeper Fielding and soon after his inswinging corner met the head of Jonathan Walters, but the Irishman's angled effort just cleared the crossbar.
And it was another Pennant corner in the 80th minute that produced Stoke's second goal, Huth's volley from 10 yards squirming underneath Fielding.
Derby manager Nigel Clough:
"Stoke are a very difficult side to create chances against.
"When they went up to Anfield a couple of weeks ago they shut up shop and Liverpool struggled to create chances against them, and we did that at times today.
"But the biggest thing we're aggrieved about is their first goal after five minutes with a player [Jerome] in an offside position."
Stoke boss Tony Pulis:
"It was always going to be a tough game as Derby have been on a great run of late.
"We started well, scored the goal, but then I thought we were sloppy after that. In the last 20 minutes of the first half Derby played really well and caused us problems.
"At half-time we dusted a few down and they went out a little more focused in the second half."