Derby defeated the Red Devils 8-7 on penalties at Old Trafford after the third round match ended in a 2-2 after 90 minutes.
Mourinho was frustrated at the end of the game but Lampard revealed he was humbled as he congratulated Derby’s performance.
“He [Mourinho] said what a great performance it was from the [Derby] team. I appreciate that because it’s not easy in defeat in that last moment.” Metro UK quoted him as saying.
“I think he actually said that before the penalties as well, but I appreciate it.
“Jose was very welcoming as a whole. We obviously know each other, and he came and spoke to me before, we spoke during and afterwards.
“It was an honour for me to go up against him.”
Speaking further, Lampard said, “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.
“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.”