City headed into the fixture in a decent position having netted two away goals in the 2-2 first-leg draw at the Parken Stadium seven days ago.
But they made tough work of ensuring their progression following a wasteful evening in front of goal, until Bellamy netted two late goals to avoid a nervy finale, and set up a tie against Danish champions Aalborg in the next round.
After a combative start to the encounter, in soggy and difficult conditions, City flirted with threatening on goal as striker Bellamy was twice flagged offside when looking to dance in on searching through balls.
City mustered their first shot on target in the 16th minute as Pablo Zabaleta marched in on a clever Robinho pass, but the angle did not favour the Argentine, who could only stab a weak effort into the hands of goalkeeper Jesper Christiansen.
WoodworkThe home side then went mighty close twice in as many minutes, as Robinho first saw a close-range header denied by a combination of woodwork and Christiansen, before the Brazilian's weaving run and low strike was smothered away.
Copenhagen provided little in an attacking sense during the early stages, as City again tested Christiansen, albeit tamely, with Zabaleta and Robinho strikes from the edge of the area easily kept out by the Danish shot-stopper.
Dame N'Doye tried his luck for a penalty under a challenge from Richard Dunne before the break, but referee Selcuk Dereli was unconvinced and waved away appeals. Thomas Kristensen was then yellow carded for hauling down Shaun Wright-Phillips.
City came within millimetres of breaking the deadlock prior to the interval after a quick one-two between Bellamy and Robinho saw the former clip an effort past the exposed keeper, only to see the ball spin back out off the far post.
Copenhagen began the second period with a change to their starting XI, with Libor Sionko replacing Kristensen, who picked up a knock prior to the break, and they went close on 48 minutes with Ailton Almeida bending a shot over Shay Given's goal.
OpenerMark Hughes' men continued to create in the second half, with Christiansen palming away a Bellamy free-kick, and then Wright-Phillips spurning an excellent opportunity as the winger opted to try and find the onrushing Ireland, who could not reach the skidding pass, instead of firing at goal.
Former Chelsea wideman Jesper Gronkjaer entered the fray, in place of N'Doye, on 59 minutes, before another City counter-attack was quelled when Wright-Phillips was cynically brought down by Oscar Wendt, who was booked for the challenge.
In the 73rd minute, to the huge relief of the increasingly cagey home support, City finally broke the deadlock as Bellamy raced onto a long Zabaleta ball, with defender Mathias Jorgensen flagging in his attempt to cut out the pass, before coolly dispatching home.
Copenhagen boss Stale Solbakken brought on Martin Vingaard in a bid to turn his side's fortunes around, but City continued to pour forward, and, after Robinho was guilty of a glaring miss that hit the foot of the post, Bellamy dispatched with aplomb from inside the box following silky skill from the aforementioned Brazilian.
The visitors, though, refused to go down without a fight and they netted a consolation in injury-time, with substitute Vingaard firing a low strike beyond the reach of Given. But it was too little, too late for Copenhagen.
1st Half Goals
Shots on Target
Shots off Target