The Spain interntional was on target from the penalty spot, with Adrian Lopez and Arda Turan also finding the net as the Liga leaders set up a tantalising tournament decider against city rivals Real in Lisbon next month.
Costa feels the fact his side did not panic when Fernando Torres gave Chelsea the lead was crucial to their ensuing success, and felt their coach was key in that regard.
"We gave absolutely everything to get to this final. We remained calm when Fernando Torres scored because we knew that with just one goal we could get through," the striker was quoted as saying by Canal Plus.
"This is something amazing and historic. We have done everything we can to be in this final.
"I think the team is doing things right but Simeone has succeeded in taking the best decisions at every moment."
Leading up to his spot-kick, Costa was cautioned for roughing up the penalty spot, but he claimed he could not get the ball to sit properly on it, before converting past Blues goalkeeper Mark Schwarzer.
"The ball kept leaning forward and I wanted it to be level, but it went well."
He also confirmed his substitution on 76 minutes was not due to any serious injury.
"The injury is just a knock, that's all."