© PA Photos
The visitors made a strong start to the match, with Romelu Lukaku having several early attempts at the City goal.
However, Boaz Myhill almost gifted the home side the lead in the 21st minute when he failed to collect a dangerous cross, allowing Jack Rodwell to take a close-range effort, but his strike went well high.
Dzeko fired City ahead in the 35th minute after Tevez picked the Bosnian out on the edge of the box, allowing him to find the top corner of the net with a volley.
The hosts had several chances to double their lead before the half-time interval, as Dzeko forced the Baggies to concede a corner before Milner was denied by the post; Rodwell saw two efforts blocked.
Once again Mancini's side gradually found their way back into the match, with Rodwell and Milner both testing Myhill, but the West Brom keeper was able to keep the deficit to just one goal.
Another strong spell for the visitors saw Hart prevent a Rosenberg equaliser, before substitute Peter Odemwingie won a corner with a 30-yard strike.
City looked to finish the game off inside the final 10 minutes, as Dzeko had a close-range effort blocked before Maicon blasted a Milner cross high and wide of the net, while Gareth Barry also called Myhill into action late on, but the hosts could not find a second goal.