Klopp and Milner were involved in a minor tiff on the touchline when Klopp sent on Curtis Jones, but a minute later they were all smiles as Jones was involved in the build-up to Mohamed Salah's opening goal.
It was easy to understand Klopp's concern given the amount of injuries to key players this season -- namely in defence with Virgil van Dijk and Joe Gomez long-term absentees and now Joel Matip also unavailable.
"My mood is constantly concerned about injuries ... and we try to save everybody, like it was with Millie as well," Klopp said.
"But as far as I know in the moment, today nothing happened injury-wise. We played a good game and got three points, which was absolutely great."
Klopp explained that he took Milner off because he was running too much.
"So the thing is, in half-time we changed the formation, offensive formation, so the set-up. Millie was now slightly deeper. We told Millie, because he had this hamstring [that] stiffened up slightly, so we told him, 'Be controlled, more passing than running with the ball' and stuff like this.
"Then I saw the last long, long sprint Millie had and I thought, 'Oh, we missed the point, we missed the moment' and I was really, really worried.
"Then he came off and said, 'You told me I had to play deeper, now I run less and now you take me off.' Then he left and came back and said, 'Yeah, by the way, good point, good decision!' All fine."
"If you want to win the league, you have to win every game," Salah, who scored twice having not registered in the league since mid-December, said. "This year City were struggling in the beginning and now we were struggling three or four games ago.
"So, we just need to carry on and keep winning."