Klopp guided Liverpool to last season's Champions League final, where they lost to Real Madrid, and also finished runner-up in the competition with Borussia Dortmund in 2013.
He also came out on the losing side in two German Cup finals, as well as a Europa League and League Cup final with Liverpool.
Goals from Mohamed Salah and Divock Origi helped Liverpool win their sixth top European title and earned Klopp his first major trophy since 2012.
"I feel mostly relief to be honest, for my family actually," Klopp told reporters at the Wanda Metropolitano stadium in Madrid.
"We always go on holiday the last six times, with a silver medal, it didn't feel too cool. It's different this year, it's for them.
"(My players and I) were pretty much crying on the pitch because it was so emotional, so big. I know how Tottenham feels in the moment better than anyone else in the world.
"They had a sensational season and would have deserved it, but we scored the goals in the right moment. I told Pochettino after the game they should be really proud of what they did."
Klopp said Liverpool, who took the battle for the Premier League title against Manchester City down to the final match of the season before eventually coming up short by one point, were just getting started.
"It is important that people don't ask us all the time about not winning," added Klopp.
"We want to win things too, 100%, this is only the start for this group. They have their best years ahead of them.
"I have a lot of silver medals and now I have a gold one, but I'm really happy for other people."
(Editing by Peter Rutherford)
