With no extra time in this season's competition, the contest went straight to spot kicks and Frank Lampard's Derby won 8-7 with Phil Jones's effort saved by visiting keeper Scott Carson.
United manager Jose Mourinho said he felt his team were in difficulty once the shootout got down to his defenders.
"I knew that we were going to be in trouble, with Jones with Eric (Bailly) I knew that we were going to be in trouble then," he said.
For former England midfielder Lampard, who played under Mourinho at Chelsea, it was just reward for playing a full-strength side and approaching the game in positive fashion.
"I wanted to bring a team who could really compete. I am so proud of them," said Lampard, in his first season as a manager.
"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," he added.
Derby had been moments away from winning 2-1 in normal time after United keeper Sergio Romero was sent off in the 67th minute but Marouane Fellaini got a stoppage-time equaliser.