Jesper Christiansen's howler gifted City their opening goal after 29 minutes as the Copenhagen goalkeeper allowed Nedum Onuoha's weak shot to slip through his fingers and over the line.
City continued to press forward but Copenhagen equalised against the run of play on 56 minutes when Ailton capitalised on poor defending to head in William Kvist's corner.
Stephen Ireland restored City's advantage five minutes later with a fine finish from Shaun Wright-Phillips' cross.
However, Martin Vingaard nodded home Dame N'Doye's cross in stoppage time to leave City with work to do in the return leg at Eastlands later this month.
PressureMark Hughes' side started brightly and carved out a good chance in the 11th minute when Ireland split the defence with a superb ball.
Wayne Bridge latched on to it and delivered the cross but Craig Bellamy had strayed offside and the move broke down.
The Danes responded and Micah Richards did well under pressure to clear a corner from Oscar Wendt at the back post four minutes later.
Then Wright-Phillips made space down the right but his curling cross was easily cut out.
The winger again got clear in the 20th minute and tried to pick out Robinho only for the ball to go behind the Brazilian.
Robinho repaid the favour two minutes later by setting up Wright-Phillips but he saw his shot blocked by Christiansen.
Copenhagen were looking dangerous from set-pieces and Mikael Antonsson got clear of his marker in the 27th minute but headed over the bar following a corner from Wendt.
City took the lead two minutes later, though, and chalked up a vital away goal thanks to a dreadful error from Christiansen, who allowed a weak shot from Onuoha to squirm under his body to the delight of the travelling fans.
The visitors began the second half brightly and Wright-Phillips skipped away in the 47th minute only to send his shot into the goalkeeper's body.
Play switched down to the other end and Zdenek Pospech carved out an opening for himself but dragged his effort wide of the post.
It was a brief respite for the home side, with Ireland cracking in a shot in the 52nd minute that was deflected to safety.
Vincent Kompany then sent Bellamy scampering clear down the left but his delivery was blocked.
Goal out of nothingCopenhagen hauled themselves back into the game in the 56th minute by scoring the equaliser.
Almeida was the first to react when Thomas Kristensen delivered the corner and sent a header beyond goalkeeper Shay Given.
However, City regained the lead in the 61st minute with another goal out of nothing. A cross from Wright-Phillips went through the legs of Atiba Hutchinson and direct to Ireland.
Such is the midfielder's confidence he was not going to waste the chance and he rattled a shot into the corner of the net past a stranded Christiansen.
Substitute Jesper Gronkjaer almost pulled Copenhagen level for a second time in the 81st minute but Richard Dunne came to the rescue and cleared off the line.
City then launched a swift counter-attack and Robinho should have extended their lead but showed a distinct lack of confidence to deliver a shot into the goalkeeper's body from a good position.
Copenhagen equalised again when substitutes Vingaard and Dame N'Doye - both on their debuts - linked up in stoppage time for the former to head home.
1st Half Goals
Shots on Target
Shots off Target