Pep Guardiola's men had won each of their first three games in Group G, including a 5-0 victory at home against Copenhagen last week, and they knew that a victory against the Danish champions would see them progress to the knockout rounds.
However, the Citizens were frustrated during an action-packed first half, which saw a wonder strike from Rodri disallowed, Sergio Gomez sent off and Riyad Mahrez miss from the penalty spot.
Nevertheless, City remain top of Group G with 10 points and they will qualify for the knockout rounds if Borussia Dortmund avoid defeat against Sevilla later this evening.
Guardiola's side unsurprisingly dominated proceedings in the opening exchanges and they thought they had broken the deadlock in the 11th minute when Rodri rifled a glorious strike into the top-left corner from around 25 yards out.
However, the Spaniard's spectacular strike was disallowed after referee Artur Dias deemed that Mahrez had deliberately handled the ball in the build-up following a pitchside review on the VAR monitor.
VAR were then involved again in the 24th minute but this time in City's favour as the referee awarded the Citizens a penalty after the ball bounced off the arm of Nicolai Boilesen when marking Manuel Akanji from a corner.
With goal machine Erling Braut Haaland named on the substitutes' bench for the first time this season, Mahrez stepped up from 12 yards but saw his shot saved by Copenhagen goalkeeper and former Liverpool youngster Kamil Grabara, who guessed correctly and dived to his right to tip the ball away from danger.
Man City's were then dealt another blow on the half-hour mark as Sergio Gomez was shown a straight red card after brining down Hakon Haraldsson as the last man on the edge of the area.
The challenge was initially waved away by the on-field referee but VAR intervened again and City's Spanish left-back was eventually given his marching orders.
Both sides fashioned half chances during a scrappy second half, but neither team were able to make a breakthrough and they were eventually forced to share the spoils at Parken Stadium.
Copenhagen will view this result as a moral victory following their five-goal drubbing against the Premier League champions last week.
Indeed, the Danish outfit have become the first team across all competitions to prevent Man City from scoring this season, ending their impressive 13-game streak.
Guardiola and co will now turn their attention to Sunday's mouth-watering Premier League clash away at Anfield against an out-of-sorts Liverpool side, who will be seeking to bounce back from a 3-2 defeat against league leaders Arsenal last weekend.