Branislav Ivanovic's fierce extra-time header earned Chelsea a 2-1 aggregate win over their Premier League rivals and booked a spot in the Wembley showpiece in March.
However, the talking points of the match centred on striker Costa, who was involved in stamping controversies in both halves with Emre Can and Martin Skrtel.
Costa also saw legitimate claims for a penalty waved away in the first half by referee Michael Oliver when he was fouled by Skrtel in the penalty area.
"I don't want to speak about decisions and decisions. If I have to speak about something I have to speak about something I know," the Chelsea boss told Sky Sports.
"When a player stops a counter-attack with a handball it's a yellow card, when a player kicks a player in the box it is a penalty. Speak about Costa and the penalty he [should have] had, let Costa play his football.
"If I speak, people will say Mourinho again. If I speak the FA [Football Association] will try to punish me like they always do, or try.
"I think it's better for you to do it. But it's so clear that even winning the game I have to ask myself why. But the most important thing is that we won, the group is happy and the supporters were absolutely wonderful.
"The final at Wembley always means a lot. Wembley has history and the fans they want to go, they pushed us."
The Portuguese, whose first trophy in English football came by beating Liverpool in the 2005 League Cup final, also paid tribute to the quality of Chelsea's opponents.
"This is a new Liverpool team and a very difficult opponent. So I'm even happier because we beat a very good team over two legs," he added.
"The difference was the goal that should have happened before in the 90 minutes. In the second half we had a lot of football."
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