Mourinho's men had to weather a second-half storm to go through 5-3 on aggregate even though Galatasaray scored three second-half goals in quick succession to win the second leg 3-2.
"I had no time to be nervous," Real's coach told reporters. "I don't have time to feel nervous on the sidelines.
"I know Galatasaray are strong. They did not play with 11 men today, they played with 50,000 incredible supporters. It was hard for us and they deserved to win tonight."
Master tactician Mourinho, bidding to become the first coach to lift Europe's top club trophy with three teams after winning with Porto in 2004 and Inter Milan in 2010, said he adjusted his team when they fell 3-1 behind.
"I put Raul Albiol in front of the defence and things started to calm down," the Portuguese explained.
"In this kind of match we had everything to lose and nothing to gain. I have always felt these players are ready to work and have great motivation to get to the finish line and play in the final."
Two-goal marksman Cristiano Ronaldo said Real would learn from their Istanbul experience.
"This match will serve as an example for us in the future," he told Spanish broadcaster TVE. "In football you cannot just give chances away.
"They made things hard but in reality we were the ones who made things complicated for ourselves. We must make sure that we are switched on for the full 90 minutes."
Galatasaray coach Fatih Terim said he was proud of the effort his players gave after appearing to be down and out following a 3-0 first-leg defeat in Spain.
"Maybe we couldn't knock Real Madrid out of the competition but we deserve the respect of the whole world," said Terim.
"I hope they win the Champions League so then we can say we lost to the champions.
"Thanks to Real Madrid and Jose Mourinho. The coach of one of the best teams in the world came to our dressing room to congratulate us," he added.
"There's nothing that Turkish players can't achieve with proper coaching. We need to improve every minute, we need to keep our heads up and continue on our way."
