The Rams fell behind to a Moussa Dembele brace at Craven Cottage, but secured their place quarter-finals with five goals in a 20-minute spell either side of the break to the delight of their manager.
"The goal just before half-time gave us a lifeline, and at half-time the players wanted to do something about it," McClaren told reporters.
"The second half was more like us, we showed good character and played some good football. It was a devastating period."
Derby will now turn their focus to Saturday's Championship clash with Brentford.