The 19-year-old midfielder, who will cost £5.1million, has signed a four-year contract but will return to former club Red Star Belgrade on loan for the remainder of the season and will rejoin the Reds in July.
Liverpool fought off interest from Russian clubs CSKA Moscow and Zenit St Petersburg, who were offering over Â£7m, following personal intervention from Klopp after being advised about the under-21 international by his Serbian assistant Zeljko Buvac.
"I'm very excited and so proud to be here and a member of Liverpool Football Club. I can't wait to get started, play at Anfield and train at Melwood," Grujic told liverpoolfc.com.
"I was thinking about my progress in my next period and decided it was good for me to come here and try to be the best I can.
"I want to play in front of one of the loudest set of fans in Europe at Anfield.
"I also like the atmospheres at other stadiums, the crowds are very good. I like to play under pressure and that's why I've come here - because it brings out the best in me."
Klopp was alerted to Grujic's potential soon after arriving as Brendan Rodgers' replacement in October and now the deal has been wrapped up.
"We're very pleased because Marko is a big talent and we've seen him a lot of times," said the German.
"When I came here, our scouts showed me some footage of a very skilled player from Red Star Belgrade.
"We watched it, we spoke to him, we met each other - he's a good boy, a young boy but plays an important role at the moment at Red Star, who are the best team in Serbia.
"To describe him, I would say he's a tall boy, but quick and good at technical things. He can play passes and can dribble.
"But he is young, so he has to develop. He will do this, so everything is okay."