The home side made much of the early running, with Craig Bellamy and Robinho posing plenty of problems for the visitors.
However, chances were hard to come by until 20 minutes in when Bellamy scampered away down the right and his deflected cross to the far post was headed against the woodwork by Robinho.
Soon after, Robinho tricked his way clear in the area, but Jesper Christiansen made a fine block at his near post.
City continued to call the shots on a rain-soaked Eastlands pitch, but the tie remained evenly balanced.
Just past the half-hour mark Pablo Zabaleta's deflected 20-yarder was well gathered by Christiansen and the keeper then collected a long-range drive from Robinho.
Right on the stroke of half-time Bellamy got clear on the left, but his shot across Christiansen came back off the far post.
Five minutes into the second half Christiansen pushed away a 25-yard free-kick from the impressive Bellamy.
A quick break then seemed certain to bring City the breakthrough, but Shaun Wright-Phillips' ball across the face of goal was inches too far in front of Stephen Ireland.
Seventeen minutes from time City finally took the lead on the night as Bellamy raced onto a long ball out of defence from Zabaleta and drilled a low, right-foot shot under the advancing Christiansen.
Robinho missed a virtual open goal soon afterwards, as City continued to call all the shots in the driving rain.
The star duo of Robinho and Bellamy then combined to make the score 2-0 ten minutes from time, as the Brazilian skipped clear on the left and played in a perfect pass for Bellamy to rifle home from ten yards out.
Martin Vingaard slotted home a consolation for the visitors with almost the last kick of the game, but City were worthy winners on the night and in the tie overall.