The match eventually went to penalties, with Mason Mount missing the deciding spotkick to see Spurs progress, despite being on the backfoot for the first hour of the game.
During the game, Mourinho and Lampard could be seen having a touchline spat, with the Chelsea boss unhappy about how much his colleague speaks to the referee and the fourth official.
Both managers played down any argument afterwards, and Mourinho even gave Lampard a piece of managerial advice, telling him to be more vocal when his team is losing, like against West Brom at the weekend.
"The only thing I was telling him, just an opinion of an old coach to a young talented coach, which was when the players need us it is when they are losing, when they are winning we don't need to be the protagonists of the touchline, we need to be there when they are losing," the Portuguese revealed after the League Cup win for Spurs.
"In the last match when they were losing 3-0 I felt really sorry for him because he was sad and quiet in his chair. In terms of football I can't teach him anything, he knows football, it is just an opinion, stay on the touchline when your team is losing and stay calm when your team is winning."
Mourinho and Lampard enjoyed huge success together as manager and player at Chelsea, and the respect remains strong between the pair.
"I owe him everything he gave me, he always gave me everything he had as a player and I never forget that," Mourinho said.
"My feelings towards him will always be feelings of how much I owe him for how much of an incredible player, friend he was."
In a bizarre moment in the match, Eric Dier could be seen sprinting down the tunnel, with Mourinho subsequently rushing in after him.
It turned out Dier simply needed the bathroom, with his boss saying afterwards: "He had to go!"