Second-half goals from John Obi Mikel and Oscar sent the Blues through to the fourth round but it was not a disappointing performance from the Rams according to their proud manager.
"(I) couldn't be more proud of them. We played against one of the best teams, not just in the Premier League, but in Europe,” said McClaren.
"(We) started the game well, finished the game well, just that five-minute spell, with the two goals, were the disappointment of the afternoon."
Of his Rams side, who have won eight of their past 10 league outings, McClaren added: "The gap is getting bigger, but teams, (but) you worry about it when you get there.
"You saw today that the difference in level is huge. Their front four are quality players, but they pay a tidy sum for them.
"We're just getting young players and developing them."
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