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.
Zinchenko, who started in midfield for Pep Guardiola's side, as opposed to his more usual left-back slot, hailed the achievement, but warned that there are still many hurdles for City to clear.
The Ukrainian international 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."
Watch the full interview with Zinchenko via the video at the top of this page.