A Branislav Ivanovic header in the fourth minute of extra-time edged a frenetic, ill-tempered clash with Liverpool 1-0 as the Blues advanced to the March 1 final 2-1 on aggregate.
Costa was involved in three of the game's main talking points - stamps on Emre Can and Martin Skrtel - and a Chelsea penalty appeal when tripped by Skrtel.
Mourinho railed against Costa's critics, one unnamed television pundit in particular, who is understood to be former Liverpool midfielder Jamie Redknapp.
"I don't know what you understand by stamp," Mourinho said.
"I think maybe you are already influenced by..
I'm going to use a word which put me in trouble but I think this time I cannot be punished to say that there is a campaign on the television with a certain pundit that is saying Diego Costa 'crimes'
This guy must be nuts.
"I saw the incidents
About the penalty, I don't speak
I prefer not to speak
If I comment I will be in trouble and I don't want to be.
"What you call stamps and Sky calls crimes, I have to say absolutely accidental
He goes to the ball, he chases the ball, as the opponent is on the floor, they have a contact, he puts his foot there when he's looking to the ball.
We know how much that pundit loves Chelsea and particularly loves me.
"When you are there and you are paid and you are very well paid - much more than some managers that have to put their ass, every 90 minutes, every weekend on the bench.
"These guys, they have a very good seat, very good money, no pressure
They are always right
They never lose, they always win, but they have to be fair and they have to be honest."
Mourinho declined to name names, adding: "Forget it
Let's go to Wembley
"I don't know his name, because when I see him I switch off the television."
Costa could face retrospective punishment from the Football Association for the stamps, particularly the one on Can which came in front of the dugouts.
Liverpool boss Brendan Rodgers said: "He's an outstanding player, an outstanding footballer.
"He has this edge to him
With Emre Can, that was poor, I felt
He could easily hurdle over the young player and he directs his studs right on to his ankle, which could've been nasty.
"And again the one with Martin Skrtel
There's no need to do it
That's the sad thing of it.
"He's a top-class player and he's clever enough that the officials don't see it.
"It was poor from him
He doesn't need to do it
It wasn't good for the game.
"The referee didn't see it
When you see it afterwards, it's disappointing."
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.
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.
Source : PA