While the Blues would strike first through Timo Werner, Erik Lamela would score a late equaliser for Tottenham before Mason Mount missed the decisive spot-kick in the penalty shoot-out.
In an entertaining match off the pitch, which saw Eric Dier forced into a mid-game toilet break, Lampard and Mourinho were also seen in a fiery exchange on the sidelines.
Lampard, who previously played under Mourinho at Stamford Bridge, was quick to stress they remain close but admitted he couldn't resist taking a jab at the Spurs manager's habit for talking too much with the officials.
"We were having some fun on the line. I get on well with Jose and I think he said that to me," Lampard said post-match.
"I commented on the fact that he seemed to speak more with the referee than he speaks with his players so it is just a moment with us.
"I have a huge respect for Jose, no matter what he says on the touchline. I think he was feeling the sense of the first half and how it was. All was well at the end of the game and that will never change."
Mourinho himself also revealed post-match that he had some touchline advice for the Blues boss, who he believes needed to be more vocal when his side was losing against West Brom in the Premier League over the weekend.
"With Frank the most important thing is my feelings towards him rather than any words we exchanged," Mourinho said.
"I owe him everything he gave me, he always gave me everything he had as a player and I never forget that. My feelings towards him will always be feelings of how much I owe him for how much of an incredible player, friend he was.
"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.
"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."