Derby won 8-7 on penalties after the game had ended 2-2 after 90 minutes. It had taken a last-gasp header from United substitute Marouane Fellaini to take the contest into a shootout.
Lampard, who played under Mourinho at Chelsea, was rewarded for naming a full-strength side, in contrast to many managers in this competition, fielding the same team that beat Brentford 3-1 in the Championship on Saturday.
"It was the easiest selection I've had. They played so well against Brentford and I knew how much they all wanted to play. I wanted to come here and win," Lampard told reporters.
It was a tense end to the game but Derby kept their cool, scoring all of their spotkicks in front of United's fans in the Stretford End.
"I’m a bit shell shocked (but) in a good way," said Lampard.
"What a performance from the team and in terms of personality...
"To have the personality to step up and take penalties and perform the way they did, I’m just a proud manager, they were great," he added.
"I am trying to build a group to be successful and nights like this are incredible. To go against world-class players and play that way, it's right up there."
The highlight of the performance was Harry Wilson's 59th-minute free kick, which was brilliantly driven home from 30 metres out by the 21-year-old Wales international, who is on loan from Liverpool.
"I know we are a good team, we wanted to come here and go toe-to-toe with United," said Wilson.
"The free kick was too far out to curl it so I put my laces through it and tried to get movement on the ball."
(Reporting by Simon Evans; Editing by Toby Davis)
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