Kevin De Bruyne got the ball rolling for Pep Guardiola's side, before the treble holders were stunned by Magnus Mattsson's eye-catching equaliser shortly after the half-hour mark.
However, Bernardo Silva restored the lead just before half time, and Phil Foden made sure of the win in second-half stoppage time, handing Man City a two-goal lead heading into the second leg.
In doing so, Foden ensured that Man City became the first team in Champions League history to score at least three goals in seven consecutive matches in the competition.
What makes the achievement even more interesting is that the Citizens scored exactly three goals in all seven matches.
City managed the feat in all six of their group games earlier this season, starting off with three successive 3-1 wins against Red Star Belgrade, RB Leipzig and Young Boys.
Guardiola's men then put Young Boys to the sword again with a 3-0 victory on matchday four, before rounding off the group with back-to-back 3-2 wins over Leipzig and Red Star.
Tonight's win puts them in a very comfortable position heading into the second leg next month, where they will now look to extend an English record that has also been set this evening.
City have become the first English side in the competition's history to win nine straight games, stretching back to their semi-final second leg victory over Real Madrid last season.
Beating Inter Milan in the showpiece occasion in Istanbul, and then completing a 100% record in the group stage gave City the opportunity to break that record, and the win at Parken has confirmed that.
De Bruyne was one of the key catalysts
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