Liverpool made Klavan their fourth signing of the summer this week.
Klavan became Klopp's fourth signing of the summer when he joined from Augsburg in a £4.2 million deal earlier this week.
The Reds had already brought in Schalke defender Joel Matip, with the centre-back signing a pre-contract agreement with the club midway through last season before officially joining at the start of pre-season.
Klavan brings with him a wealth of experience, and Klopp admitted that the 30-year-old was a player who had impressed him even while the manager was in charge at Borussia Dortmund.
The 49-year-old admitted that Klavan's impressive performances for Augsburg against Liverpool in the Europa League last season were the final confirmation that the Estonia international was a man he wanted to bring to Anfield.
Klopp told liverpoolfc.com: "At the end of last season I knew that if we decided to take somebody in this position he was the one I wanted. Now I'm really happy with the options we have got at centre-back.
"I've seen up close for three seasons how good his performances in Germany have been and last season we got a chance to see him against us - and a lot in preparation - for our games against Augsburg.
"He will bring us really good qualities with his defensive intelligence and ability on the ball and he is a left-footed player so will give us extra balance. I know that he is not only a very good player but the type of mentality and person who is coming here to fight and to win."
Klavan has well over 400 competitive senior appearances to his name in a career which has thus far seen him play in European countries including the Netherlands and Germany.
He will now compete with Matip, Mamadou Sakho, Dejan Lovren, and Tiago Ilori for one of Klopp's two centre-back spots this season.