Despite dominating the first-half at the Etihad, City went behind when Kylian Mbappe scored for the Parisians against the run of play early in the second half.
Ukraine international Zinchenko said: "It is a massive achievement for the club, for the team, but we still have a lot of games to play.
"The support from our fans is always amazing, doesn't matter if we are playing home or away, so I would like to say thanks a lot for that.
"Everyone is so happy but we need to switch on straight away for the next game because we have a lot of challenges ahead of us."
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