The centre-half underwent the planned procedure on his knee ligament injury in London and now turns his focus on beginning his rehabilitation with the club's medical staff.
Speaking at a press conference on Friday afternoon, Klopp welcomed the positive update regarding the No.4.
The Reds boss said: "First and foremost, I don't think it's the scenario or the place to talk about personal things with Virgil or whoever.
"What I can tell you, after a surgery – and it was successful, which is really good, the best news you can get – is that this time is always unpleasant. Everybody who had a surgery knows that.
"You don't wake up and everything is fine, you feel still the surgery. He's good, he's as good as you can be. That's it."
On what the club will do to help the 29-year-old in this period, Klopp continued: "Everything. Easy answer. Absolutely everything.
"It's all individual, it's all personal what this person needs or this person needs, but we do absolutely everything. We did always and we will do this time, 100 per cent."
Later asked whether Van Dijk's previous fitness record would count in his favour in his recovery, Klopp replied: "100 per cent. Clear to say.
"He's a quick healer and he's a pain-coper, so that's all clear. All these things, yes, they help."