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Liverpool, beaten by Arsenal on Sunday, lead the table on 51 points from 23 games but City are up to 49 from 22 and Arsenal on 49 from 23.
"Congratulations to Arsenal. We were not good enough today," Klopp said. "It just shows the boys are human beings."
Fourth-placed Aston Villa suffered a 3-1 home defeat by Newcastle United, Crystal Palace came from behind to beat visiting Sheffield United 3-2 and struggling pair Fulham and Everton fought out a goalless draw at Craven Cottage.
While Liverpool's 7,000 travelling fans celebrated, Arsenal's left the Emirates Stadium chuntering about the need for a new striker in the transfer window.
Bukayo Saka's early goal was cancelled out by Raul Jimenez before Bobby De Cordova-Reid's second-half strike ended the hosts' three-match losing streak.
Second-placed Arsenal remain on 40 points, two behind Liverpool who beat Burnley 2-0 on Tuesday.
Manchester City slipped out of the top four as Tottenham Hotspur made up ground with a third consecutive victory, beating Everton 2-1 at home.
Arsenal's second league defeat of the season leaves Mikel Arteta's men one point behind Liverpool, who went top with a controversial 2-1 win at Crystal Palace earlier on Saturday.
Arsenal have now taken nine extra points this season courtesy of goals scored in the last five minutes of games.
Anthony Gordon scored the only goal in Newcastle's deserved 1-0 victory over Manchester United that propelled them up to fifth in the table on 26 points.
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