The home side threatened early on as Cesar Santin found space down the left, but his pull-back across goal was cleared.
By now snow was falling steadily at the Parken Stadion, with the Danes continuing to make most of the running, although Wright-Phillips was starting to show up well on the right.
Midway through the first half Wright-Phillips had an opening 12 yards out, but his weak shot was blocked by keeper Jesper Christiansen.
Soon after William Kvist fired over when well-placed in the City area and just before the half-hour mark City took the lead.
Nedum Onuoha strode forward from the back and shot weakly towards goal from just inside the area, only for Christiansen to fumble the effort and allow the ball to roll into the back of the net.
Pablo Zabaleta wasted a good opening right on the stroke of half-time, dribbling a shot past the near post after running clear down the right.
Wright-Phillips tested Christiansen in the first minute of the second half, as City looked to build on the boost of holding a one-goal lead.
However, they were pegged back with 56 minutes on the clock as Kvist's left-wing corner was met by the unmarked Ailton Almeida and he powered a header past Shay Given from eight yards out.
City were back in front on the hour as Wright-Phillips played the ball in from near the right corner flag and Stephen Ireland met the cross with a perfect low finish inside Christiansen's near post.
The lively Craig Bellamy then sliced a shot wide as City began to punch holes in the home defence.
Robinho placed a shot straight at Christiansen when he was clean through eight minutes from time and the keeper then denied Wright-Phillips, before the hosts struck right at the death.
Dame N'Doye was allowed time and space on the right and his cross was headed home by the unmarked Martin Vingaard to make the final score 2-2.