The captain and his deputy were both disappointed in the Europa League round of 32 result at the WWK Arena, but moreover, were annoyed that the Reds could not turn in a sharper display against their Bundesliga opponents.
Markus Weinzierl’s men were the better side in the first half, and continued to defend and counter well in the second period.
Klopp expected Augsburg to “fight for their life” given it was the most prestigious fixture in the club’s 108-year history, but he also anticipated a more fluid showing from his charges.
Daniel Sturridge should have fired the Reds in front after great build-up play from Milner and Roberto Firmino but he could not convincingly connect with the England midfielder's ball across goal.
The visitors also had a dangerous Alberto Moreno free kick, which caused a scramble in the box, cleared away.
After watching his side only shine in brief spells, Klopp said he wants Liverpool “to play the level we are able to more often” and his leaders on the pitch agreed.
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“Both teams had good chances to score,” reflected Henderson. “We didn’t play as well as we could have, we did only in moments, but not enough in the game.
“They had a very good set-up, they sat in and defended well. They were very hard to break down, but we can play a whole lot better than what we did.
“If we moved the ball quicker we could have caused them more problems so that’s what we’ll look to do in the next leg.”
Milner added: “We’re a bit disappointed, we know we can play better than that. We showed our quality in bits of the game, but probably not enough.
“Both team had chances, but we feel unlucky that we didn’t score some good chances like Studge’s and the free kick which the keeper dropped on the line.
“It didn’t go for us tonight, but I don’t think we deserved to win as we didn’t play well enough tonight. 0-0 is not a terrible result, we’ll hopefully get the result at home.
“We’ve got a bit of break now for the first time in a very long time so we’ll be refreshed by Thursday.”