Having being beaten 3-0 at the Santiago Bernabeu last week, the Turkish champions were dealt what looked like a knockout blow when Cristiano Ronaldo netted an away goal for Los Blancos in Istanbul.
However, Galatasaray staged a sensational second-half fightback, striking three times through Emmanuel Eboue, Wesley Sneijder and Didier Drogba, and thought they had levelled matters on aggregate when the latter netted again, only to be ruled offside.
"When it went 3-1," Terim told NTV Spor, "Yes, I started to think of going through to the semifinals. It had not been the case up until that moment.
"But especially just after Drogba's disallowed goal, that was the the moment when we really believed it."
Unfortunately for the hosts, there was to be no coming back from that disappointment, and Ronaldo netted again in injury time to put the outcome beyond all doubt. Terim still had nothing but praise for his players and their supporters.
"Incredible supporters and an incredible atmosphere - it makes me happy that we played with such passion and gave them such pleasure," Terim said. "We must say thanks to [Jose] Mourinho and Real Madrid. He came to our dressing room to congratulate our players.
"I must also congragulate my players, as they've beaten one of the most powerful teams in the world.
"From the very beginning my aim was to create a Galatasaray that would honor their supporters. I see this happening today. I'm happy to have kept my promise."
Madrid now progresses to the last four where it is joined by Borussia Dortmund, who overcame Malaga 3-2 on aggregate with a late comeback of its own.