Chelsea’s record goalscorer Lampard, who is now the Rams manager, was given a good ovation before kick-off but another returning Blue made a poor start as the hosts took an early lead.
Fikayo Tomori, who is on loan from Chelsea, turned into his own net when he mishit a clearance after five minutes. Jack Marriott, who scored against Manchester United in the last round, responded with a quickfire equaliser but Chelsea reclaimed the lead with another own goal, this time from Richard Keogh.
Martyn Waghorn made up for an earlier miss by levelling for a second time but Cesc Fabregas swept Chelsea back ahead before the break. Derby rallied in the second half, and hit the post through David Nugent late on, but Chelsea held on. .