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Timo Werner had scored his first goal only for Erik Lamela to strike late into the second half.
The match went to penalties, with Mason Mount missing the crucial kick to see Spurs go through.
During the clash, tempers flared on the touchline between Lampard and Mourinho.
The Special One, who coached the Chelsea boss as a player at Stamford Bridge, could be seen telling him to be quiet.
It follows on from a similar situation between Lampard and Klopp, who had heated words during Liverpool's recent win over the Blues.
However, former team-mate Trevor Sinclair has praised the ex-Derby boss - who loves seeing Lampard show some passion.
“I like the fact Frank stands up for himself. I think he’s his own man,” Sinclair told talkSPORT.
“He’s got to be looked upon by his players that he cares.
"I think that’s the reaction, from a players’ point of view, I want to see my manager arguing with an opposition manager if he’s giving it loads.
“I want to see him sticking up for us like he cares for it. I think Frank’s done that.
“He’s not going for little managers because we have seen it with Klopp. He doesn’t care.
“Frank’s done everything in the game as a player.
"If he wants to emulate someone like (Carlo) Ancelotti, he has to stand his ground in the technical box and I think he’s doing that superbly well.”
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The pressure has ramped up this season in west London after a spending spree that has exceeded £200m.
Chelsea finished fourth last year after not spending a penny but with the likes of Kai Havertz, Hakim Ziyech and Ben Chilwell the expectations have grown.
But after early losses to Liverpool and Tottenham, eyebrows have been raised while also a 3-3 draw with West Brom showed the Blues still have huge defensive problems.