At this early stage of the season, the Reds boss has seen plenty of positives but knows there are areas in need of improvement, with Arsenal - also on two wins from two - up next.
"Parts of [the start to the season] I am really happy with," he told a news conference. "It's exactly the same as the beginning of last season, when we had sensational results but people asked me about the performances.
"To make the next step, first you have to reach the level of last year, the last level you had. That's what we are trying to do, but that doesn't mean after the summer break it is immediately like this.
"We know we can improve, but it's not about being perfect at the start of the season, it's about being ready, and we were. Now we work on being perfect. We may never reach it, but we work.
"Against City, second half was really good. Against Norwich, 60 minutes was really good then we obviously didn't know any more how to control a game. That was not perfect, true.
"Then we played Chelsea, in difficult circumstances but we won the game.
"Then last week, Southampton was one of the most challenging games I've had so far. The only game I can remember similar was when we came from Spain in the Europa League and played Swansea. I liked that game as well.
"Southampton had chances but that's football. We don't go into a game and think it's not allowed that the opposition shoot at our goal, that's not how defending works. That's what you have a goalie for.
"There are a lot of positive things, but a lot of space for improvement, too. It's a team sport, and you have to work. Then let's play football.
"So far, so okay, not more. Now let's go again against Arsenal."
The Premier League transfer window closed without Liverpool doing major business, but they could yet lose Dejan Lovren, who had been linked with Roma.
Klopp said he "surprisingly" had no update on Lovren, adding when pushed again later on: "Nice try. No comment."
Discussing the differing transfer windows, Klopp said: "I don't care when it closes, but it must close at the same time. That was the idea in the first place.
"They spoke about finishing it before the season starts - good idea - but only England did it. That makes no sense."