Liverpool have won all three of their league games so far this season, beating visiting Leeds United 4-3 and winning 2-0 at Chelsea, but while praising Klopp's side Keane said they had shown the ability to win while not being perfect in defending.
"Strangely enough if you look at Liverpool, they have been sloppy against Leeds, OK done well against Chelsea, sloppy again tonight, but they have got nine points, maximum points, so they will know they need to improve," noted Keane.
"It looks like they are capable of winning football matches without being at their best and that is a sign of a good team."
But Klopp pounced on Keane's comments.
"Did I hear it right, that Mr Keane said it was a sloppy performance tonight?” asked the German coach as he began a post-match interview with Sky Sports. “I could hear you already, did he say that?”
After Keane attempted to explain his remarks, Klopp continued.
"I just wanted to hear it. I’m not sure I heard it right, maybe he was speaking about another game... it cannot be this game, sorry!
“That is an incredible description of this game, that was absolutely exceptional. Nothing was sloppy, absolutely nothing," said Klopp, to the delight of Liverpool fans on social media, who relished the strong riposte to an old rival.
“Alisson had to make one save, they had two balls in behind, you cannot avoid that. You need a goalie in these situations. Apart from that, the football we played was absolutely exceptional tonight," added Klopp.
“About this game tonight, there is nothing bad to say. It was the opposite of sloppy."
Arsenal had taken the lead through Alexandre Lacazette after a poor miscontrol from Andy Robertson in his own area and the French striker could have equalised in the second half when he was one-on-one against Alisson.
Keane, known as a tough tackling player and tough talking pundit, appeared a little amused by Klopp's reaction.
"Very sensitive, Jesus. Imagine if he'd lost?" he said. (Reporting by Simon Evans; Editing by Ken Ferris)