Although Milner's goal just past the hour gave Liverpool the lead in a match that proved a stern test for both sides, Klopp admitted the midfielder had sterner words for his teammates at half-time.
An open first stanza had Klopp cutting a frustrated figure on the touchline, unhappy with the team's defensive shape. Before he could relay his thoughts, however, Milner aired his own grievances.
"You have a compact formation, you force Arsenal to pass the ball somewhere – that's pressing, it's not that complicated," Klopp explained. "The best way to do it is right from the first moment and then you can build on that, but we didn't do it right from the first moment.
"Milly was the one who saw it first and was quite – how can we say, was not happy – was animated. That's good. That's how it is in a dressing room, there are a lot of men, a lot of testosterone.
"One is talking – that's me – but sometimes before I start talking the boys have a few words and that's all. Milly is a proper leader."
The 32-year-old remains a valuable figure when Liverpool absorbs and plays on the break. Though merely eight minutes from victory in London – and despite a sarcastic tone – Klopp singled out his impact both on and off the pitch.
"Milly is like wine, a very good red one. I'm not sure if Leeds is famous for wine? He's getting better," Klopp joked.
"I think he should buy all of the newspapers and read this. I would like it if you write a story about James Milner.
"Obviously someone might write he (should) play for England again. That's a really nice idea."