The European champions qualified for Sunday's season curtain-raiser courtesy of their second-placed finish as City completed an unprecedented domestic treble in 2018-19.
They overcame Olympique Lyonnais 3-1 on Wednesday to end a run of four winless friendlies and Klopp is looking for more with the new season around the corner.
"All round it was a much better performance, but again, I said that will happen," Klopp told his club's official website.
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"Now we have three more days to prepare for the City game and then we will see what we can do there.
"I am of course happy, but I know we must play much better to have a chance against City.
"Man City is different to all other games that you play – you have to play a pre-season game against Man City to prepare for Man City and we didn't do that!
"We need to adapt to that, but all good. Today [Wednesday] was, from a physical point of view, very, very important for all the lads, and from a psychological point of view as well.
"We are all human beings and losing football games we never get used to, so I'm really happy that we could change that."
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