Pep Guardiola's men exploded out of the blocks on Nordic soil and took an early lead through Kevin De Bruyne, only for the hosts to draw level through Magnus Mattsson, the beneficiary of a horrific goalkeeping gaffe from Ederson.
However, Bernardo Silva restored City's lead just before the break, and an injury-time Phil Foden effort means that the treble winners will take a healthy two-goal lead into second leg at the Etihad on March 6.
A handful of changes on the visitors' end saw Jack Grealish handed a rare start - his first since the FA Cup beating of Huddersfield Town on January 7 - while highly-rated Copenhagen attacker Roony Bardghji was overlooked for a place in the hosts XI.
Copenhagen's Parken base has been an intimidating ground for many a European team - including Man City, who could only manage a draw there in the 2022-23 group phase - but the holders were immediately in the ascendancy in Scandinavia.
With just four minutes gone, Nathan Ake sent in a delicate delivery for De Bruyne - who failed to connect properly with his wayward header - but the Belgian quickly made amends just seven minutes later.
Cutting inside from the right flank, Foden took three Copenhagen players out of the game with an exquisite pass to De Bruyne, who picked out the far corner with a terrific strike across goal to give Guardiola's men the lead that their domination warranted.
Delight soon turned to despair for Grealish, though, as after worryingly going down before attempting to play through the pain, Guardiola opted to withdraw the Englishman, as Jeremy Doku was sent on in the 20th minute.
City's momentum was not disrupted by Grealish's blow, though, and Copenhagen centre-back Denis Vavro nearly gifted the visitors their second in the 23rd minute, heading a Silva cross against his own bar.
It was seemingly only a matter of time before the net rippled again as Erling Haaland cleared the bar with an acrobatic effort on the half-hour mark, and the second goal of the contest would arrive in the 34th minute, albeit one which lifted the home fans off their seats.
A cataclysmic error from Ederson saw the Brazilian play the ball straight to Mohamed Elyounoussi, whose strike was blocked by Ruben Dias, but the ricochet fell for Mattsson, whose stunning first-time curler from the edge of the box flew just inside the post.
There was no better way for Mattsson - a January arrival from NEC Nijmegen - to endear himself to the Copenhagen faithful, but in a cruel twist of fate, the midfielder was inadvertently to blame for City's response, which arrived in the first minute of injury time.
From a similar area to where Foden played in De Bruyne for the opening goal, the Belgian's attempted pass deflected off of Mattsson and into the path of Silva, who elegantly flicked the ball over Kamil Grabara and into the side of the net.
The die was certainly not cast for Copenhagen, though, and barring a fierce drive from De Bruyne in the 54th minute - which Grabara tipped behind - City were not at their scintillating best in the opening phase of the second half.
Ederson still remained untested, but the champions were indebted to Ake for preserving their lead in the 61st minute, as the Dutchman's vital clearing header prevented Elyounoussi's cross from reaching Andreas Cornelius for a simple finish.
Excitement was at a premium as the game entered its final quarter, although Grabara had to make a fine one-handed save to parry a long-range Doku strike away in the 69th minute, and City soon ramped up their efforts to increase their advantage.
Copenhagen were living dangerously in injury time, as Haaland saw two strikes saved either side of a blocked Matheus Nunes attempt, and City's third would eventually arrive in the second additional minute, as De Bruyne and Foden played a slick one-two before the latter fired home in the centre of the box.
City now return to their Etihad headquarters for a Premier League showdown with Chelsea on Saturday evening, while Copenhagen are away to Silkeborg in the Danish Superliga on Sunday.