The 30-year-old moved to Anfield after the Reds agreed a reported £4.2m fee plus add-ons with the German outfit.
Klavan, who spent the last four years playing in the Bundesliga, is expected to provide cover, and heads to Merseyside having kept 11 clean sheets in 31 appearances last season.
"I'm really honoured to be part of this amazing club, this amazing team," Klavan told the club's official website. "It's hard to put it into words, because it's been a dream of mine for 22 years to come to the Premier League and be a part of this amazing club, Liverpool.
"The Premier League is the most attractive league in the world and you always have amazing players here. This is the home of football."