Derby defended well and kept the Blues out for the first half, threatening several times themselves. However, a rare goal from John Obi Mikel was quickly followed by another from Oscar gave José Mourinho's men a comfortable 2-0 win.
The game got off to a rather sluggish start, with Chelsea enjoying a lot of the ball but not making much progress against a diligent and hardworking Derby County. With the Blues mostly restricted to half-chances, the best opportunity of the game at that point fell to the visitors, as Craig Forsyth's drilled cross was inches away from being turned in by a lunging Jamie Ward.
Derby began to grow into the game as halftime approached, with Simon Dawkins and John Eustace both seeing decent efforts blocked or sent wide, but Chelsea showed a glimpse of their threat afterwards. Michael Essien sent Samuel Eto'o springing free, but Andre Wisdom was able to make a recovering tackle before Eto'o could get his shot away. Just before the end of the half Chelsea went close again through Ramires, but his effort was deflected off Jeff Hendrick and struck the post.
After the second half began, the game was still looking relatively open, but Chelsea found their breakthrough via a surprising source. It was John Obi Mikel with the goal, the Nigerian bagging only his fourth goal for the club after getting on the end of Willian's delivery to put the Blues in front.
Derby had been looking good up until that point, but they suffered another blow just afterwards when Oscar made it two. The Brazilian went on a run and sent in a shot from the edge of the area which Lee Grant failed to stop, letting it slip into the net.
Thereafter, the hosts crumbled, and Chelsea could have put a lot more gloss on the scoreline afterwards. Fernando Torres rounded Grant but somehow failed to score, and Ramires hit the woodwork once again. The hosts did not threaten again and the result was a routine win for Mourinho's men, where they will be clear favourites to overwhelm Stoke City in the fourth round.
Derby County: Grant, Wisdom, Buxton, Keane, Forsyth, Eustace (Bennett 70'), Hughes, Hendrick, Dawkins (Bailey 82'), Ward (Sammon 70'), Martin.
Chelsea: Schwarzer; Azpilicueta, Cahill, D Luiz, Cole; Essien (Hazard 55'), Mikel; Ramires, Oscar (Baker 87'), Willian; Eto'o (Torres 64')
Goals: Mikel 66', Oscar 71'