The win against Borussia Dortmund was also Kroos' 300th for the club.
He said afterwards: "My intention was to leave like this. I've really enjoyed the last week, with such a spectacular atmosphere but I was also focused on the final. We wanted to win this match and we prepared for it during the week. The first half didn't go well, but it didn't matter because this team always keeps going and you have to score goals to win. I'm so happy to be leaving like this and there's no better way to do it.
"What I've experienced in these 10 years has been unforgettable and there will be times when I'll miss it. I've taken this decision because I want to lead a different life. I'll miss it, but I wanted to leave in this way and I'm very happy."
Kroos has played 465 games for Real Madrid and won 23 trophies. He has also scored 28 goals.