The centre-back suffered the issue when he was clattered by Everton goalkeeper Jordan Pickford during the Merseyside Derby.
A statement from the club released on Friday morning read: “Virgil van Dijk has undergone surgery on the knee ligament injury he sustained earlier this month.
“The Liverpool defender’s planned operation took place in London and was conducted successfully.
“Van Dijk will now immediately focus on the beginning of his rehabilitation with the support of the Reds’ medical department.
“No timeframe has been placed on the Dutchman’s return to action.”
The news will come as a relief to Liverpool supporters, although it is still expected Van Dijk will be missing for the foreseeable future.
Meanwhile Reds goalkeeper Alisson has been the latest player to hit out at Pickford for his reckless tackle.
“Every goalkeeper has a different way to play and a different way to move,” Alisson told Sky Sports News.
“In that moment, 100 per cent it wasn’t a necessary movement. Thinking in my mind when I go to a situation I go to finish it because if I go too soft the opponent can score.
“But everyone has their own way. I don’t agree with that way, everyone can see that. It was an unnecessary moment and movement.”
Jurgen Klopp has tended to opt for a partnership of Joe Gomez and Fabinho at centre-back since the loss of Van Dijk, and the same two players are expected to play in the Premier League game against West Ham on Saturday.