The champions ended Arsenal's 100 per cent start to the season while extending their own at Anfield to make it 61 home league matches unbeaten.
Having lost their last two matches to Mikel Arteta's side - one on penalties in the Community Shield - there was barely a time over the 90 minutes when the defending champions looked like allowing it to happen a third time.
Sadio Mane, Andy Robertson and Diogo Jota, on his home debut, all scored after Alexandre Lacazette had been gifted the opener after a Robertson error.
However, three points was not enough for Klopp, who described Liverpool's performance as "exceptional" after taking issue with Manchester United legend Keane using the word "sloppy" in his analysis.
Reds boss Klopp overheard pundit Keane giving his assessment in the Sky Sports studio as he waited to be interviewed at the side of the Anfield pitch.
🗣️ "Did Mr Keane say it was a sloppy performance? Maybe he is speaking about another game..." 👀
Jurgen Klopp v Roy Keane 🍿 pic.twitter.com/MIw47L6N0o— Sky Sports Premier League (@SkySportsPL)September 28, 2020
The German said: "Did I hear right that Mr Keane said we had a sloppy performance tonight, because I could hear you already? Did he say that?
"This was a sloppy performance tonight? Maybe he spoke about another game. It cannot be this game, sorry.
"That's an incredible description of this game. This was absolutely exceptional. Nothing was sloppy, absolutely nothing. It was, from the first second, dominant against a team in form, 100 per cent in form.
"About this game tonight, there is nothing bad to say."
Former Manchester United midfielder Keane swiftly moved to clarify his comments.
"I think you might have misheard me. I said there were sloppy moments but I think you have been outstanding," said Keane.
"I've been giving the club nothing but praise. I'm not sure you heard me correctly there."
Additional reporting by PA.