The Reds made a blistering start to the match, going ahead thanks to a Fikayo Tomori own goal and should have been 2-0 up as Mohamed Salah missed a penalty.
Somehow they found themselves 2-1 down at half-time with Milan scoring through Ante Rebic and Brahim Diaz in the space of two minutes.
Virgil van Dijk was on the bench for the game and Klopp will be disappointed with the goals his side conceded.
They went on to win 3-2 thanks to a brilliant Jordan Henderson goal after Salah had equalised.
talkSPORT host Jamie O'Hara believes Liverpool made hard work of the result and pointed to the role Trent Alexander-Arnold played in the goals.
He told The Sports Bar: "For me, Trent was at fault for both goals. Positionally, he's all over the place. I think he forgets he plays right-back.
"I think he's amazing going forward, he's got so much talent and I love watching him play when he's got the ball at his feet, but when he hasn't got the ball he is so bad at getting back.
"He gets done with the one-two and gets out of position.
"First and foremost, you are a right-back and have to defend. You are playing a big team and without Van Dijk, they got exposed.
"They won the game, but they made it difficult. Against better opposition in the Premier League they will get done."
Co-host Jason Cundy added: "I think that has got to come from Klopp. His role, and Robertson on the other side, they play like wingers.
"That's the way they play. Defensively, I'm not so sure. A couple of times in the second half Salah, when Trent overlapped, to his credit he got back.
"I wonder if Klopp has said to him that if Trent goes beyond you, your job is to cover him.
"If that's what he's told to do, then I feel sorry for him.
"He's so good at it. He's essentially a winger when Liverpool have got the ball in the high and wide areas.
"He creates so many chances and they have to legislate for that. I'm wondering if the midfield has got to take some responsibility.
"For the first 25 minutes they were truly awesome, but they went into half-time 2-1 down. That will concern Klopp.
"They were so dominant yet they went into half-time 2-1 down."
The Reds picked up all three points in their opening group game, but there was plenty for Klopp to ponder.