The Rams came from behind to produce a memorable performance en route to a penalty shootout win, progressing through to the fourth round of the EFL Cup at United's - and Lampard's former boss Jose Mourinho's - expense.
The highlight of the win was a stunning free kick from Liverpool loanee Harry Wilson, but Lampard expressed his pride in all of his team having overcome a third-minute goal and then a 95th-minute equaliser from the hosts.
"I'm shell-shocked. What a performance. To go down early at Old Trafford and play as well as they did was superb. We had the personality to take penalties and play the way we did. I am a proud manager," Lampard told Sky Sports News.
"It was the easiest selection I have had, the players were superb against Brentford and they deserved the chance to play at Old Trafford. I wanted to bring a team who could really compete. I am so proud of them.
"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. Harry Wilson does do that in training, he proved me wrong as I thought it was too far out. Both he and Mason Mount are going to go a long way."
The victory was Derby's first at Old Trafford since 2001.