Late goals from Dani Carvajal and Vinicius Jr saw Real win 2-0 at Wembley against Borussia Dortmund.
Terzic said: "Difficult. Very tough to find the right words. Performance wise we showed a great game but we also found out why they (Madrid) just became European champions for the 15th time. They have been so effective and this is something we missed. The proud thing is that we gave a really good game and we showed everyone that we came here not just to play a game but to win.
"We were very close and it's been small things missing. Congratulations to them. To keep this kind of hunger to go and win it again, you see why they are the true champions.
"Yes I am. That is the first thought. A proud moment. We brought nearly 100,000 people to London and everybody had the belief we could make it. It was a fantastic journey this year in the Champions League a great campaign, but to be honest it is a little empty inside. It was a great opportunity to win it and we unfortunately didn't make it."
He added: "It is on us to push on and keep the belief high. In life everything is about belief and if you keep the belief high then comes the trust. In the world of sport there is always a winner and a loser and we were close to it but it's not only to taste the feeling but keep the belief high. No room for doubts."