Goals from Sadio Mane and Roberto Firmino ensured Liverpool kept their 100 per cent League record this season despite a second-half blunder from goalkeeper Alisson.
The win sent Liverpool clear at the top of the table, with Klopp's men now nine points ahead of United after the Red Devils' poor start to the season.
Mourinho hit out at his critics in the build-up to the weekend's action, claiming that Klopp and Tottenham boss Mauricio Pochettino are not subjected to the same kind of pressure as him despite neither manager winning a trophy in England yet.
The Portuguese coach took further aim at Klopp's trophy record by bringing up the fact that the German is yet to win any silverware outside his home country.
When asked about Mourinho's assertion after Liverpool's win at, Klopp replied: "What? What did he say? Now I’m interested."
After having Mourinho's comments explained further, Klopp said: "He’s right, that’s right. If he is speaking about me. Probably there are a few more managers in the league but you thought it was about me – no problem.
"He is right absolutely. He is probably the most successful manager in the [Premier League] in that moment. I have no problem with that.
"I don’t watch Jose Mourinho press conferences - should I? I will start doing that. I cannot say anything about that.
"[I have] no clue what you are talking about to be 100 per cent honest. If he said I didn’t win anything outside Germany then he’s right - that’s easy to see."