The 19-year-old midfielder, who will cost £5.1m (€7m), 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 more than £7m (€9.5m), following a personal intervention from Klopp after being advised about the U21 international by his Serbian assistant Zeljko Buvac.
Klopp was alerted to Grujic’s potential soon after arriving as Brendan Rodgers’ replacement in October and now the deal has been wrapped up.
“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.”
While the arrival of Grujic marks the start of Klopp’s restructuring of the squad he inherited from Brendan Rodgers the German has more pressing concerns which need to be addressed in the January window.
Injuries to centre-backs Martin Skrtel, Dejan Lovren, Mamadou Sakho, and Joe Gomez mean he is looking for cover with a busy month ahead.
Liverpool do have one other centre-back in Tiago Ilori who is on loan at Aston Villa but he has yet to make an appearance for the Midlanders — or indeed his parent club, having arrived from Sporting Lisbon in September 2013.
Liverpool are tracking Schalke defender Joel Matip, who is out of contract at the end of the season, but there remains plenty of work to be done to sign the German-born Cameroon international.
Meanwhile, midfielder Joe Allen admits he is playing for his long-term future at Liverpool. The Wales international had made just one Premier League start this season — at Newcastle a month ago — and with contract discussions which began last March on hold, the 25-year-old admits he has to work hard.
“I have made only one league start this season and I want to play more. From my point of view it is about knuckling down, doing my job and giving my best.”
Allen’s performance in the 1-0 Capital One Cup semi-final first leg win at Stoke on Tuesday was arguably one of his best at Liverpool.
Allen was not the only one to raise his game at the Britannia Stadium after some angry words from Klopp after Saturday’s dismal defeat at West Ham.
“It was some frank words. As a team we needed to hear them I guess but this squad has great personalities and great character, players who want to work hard, so there is no issue there,” he added.
“We knew we needed to respond with a good performance.”
Liverpool currently have six players out with hamstring problems and Klopp’s FA Cup rival Paul Tisdale accepts playing in a certain fashion puts players at greater risk of injury.
Liverpool head to Exeter tomorrow night for his FA Cup debut.
“Maybe the propensity for certain types of injury are different,” Tisdale said.
“By definition, if you play in a certain way, the ball is played in a certain area of the field and you play with a different flow I guess there will be (injuries).
“There’s so much football with cup competitions and you expect players to train with so much intensity from Saturday to Saturday that injuries are a product of the modern game.”