Tempers frayed in the technical areas as Mourinho and Rodgers confronted each other.
Rodgers, who thought Chelsea goalkeeper Thibaut Courtois was the difference in the tie, played down the incident but Mourinho hinted at animosity towards his former protege.
Rodgers said: "There's no problem. It's a competitive game, it's a big game.
"Two top sides, so the 'chimp' comes out every now and then," he added, alluding to the club's link with sports psychiatrist Professor Steve Peters and his 'inner chimp' method.
Mourinho, who saw Courtois save well from Alberto Moreno and Philippe Coutinho in the first half, added: "We are very different people.
"He worked in Chelsea in my time for a few years. I don't know if he learned something with me or not. Maybe he did, maybe he didn't.
"I could speak about Skrtel, I don't. He goes to the television, he speaks about Courtois. He forgets (Liverpool goalkeeper Simon) Mignolet. I speak about Courtois and Mignolet.
"He forgets his goalkeeper. I don't forget my goalkeeper, because my goalkeeper helped me to be in the final.
"Different profiles. He is what he is. He is a fantastic manager, but we are different people."
Mourinho was offered Liverpool assistant boss Colin Pascoe's glasses as his complaints at the officials continued.
Mourinho was not amused and his displeasure with the officials continued into extra time as he missed Ivanovic's header while talking to fourth official Phil Dowd.
Mourinho added: "Yes. I missed the goal. I was saying that Lucas and (Jordan) Henderson should be off with two yellow cards."
Mourinho praised his side's commitment in a game which came four days after the FA Cup elimination to Bradford.
"They gave absolutely everything," said Mourinho, who spoke of a foot injury for Ivanovic.
"Ivanovic's boot should be in the lobby of the academy for the kids to see a boot completely full of blood. The white boot is completely red. And that's only one example."
He also lauded the Chelsea support, which he has criticised three times this season.
"Chelsea supporters were fantastic. The victory is for them," said Mourinho, who revealed Filipe Luis (calf) and Cesc Fabregas (hamstring) were substituted due to injury and are doubts for Saturday's Premier League encounter with Manchester City.
Chelsea will play Tottenham or Sheffield United in the final.
"I want the team that deserves to be (in the final)," Mourinho said.
Jose Mourinho jumped to the defence of Diego Costa after the striker twice appeared to stamp on opponents during Chelsea's victory over Liverpool which earned a place in the League Cup final.
An extraordinary game needed something extra. On a compelling, contentious night at the Bridge, Branislav Ivanovic headed home in the first half of extra time to send Chelsea to the League Cup final at Wembley where they will face either Sheffield United or Tottenham Hotspur.
This was a game decided on the finest margins so it was perhaps fitting that when Branislav Ivanovic headed the only goal in extra time, Jose Mourinho was too busy making a point to the fourth official to watch Chelsea take the lead.