The Citizens rotate their squad heavily in preparation for the FA Cup final at the weekend and take the win thanks to the Bosnian's first-half volley in a game filled with chances
The home side dominated the first half and took the lead after 34 minutes thanks to a superb volley from Edin Dzeko, having created a number of other chances beforehand.
The Baggies improved in the second half as the hosts seemed to lose their focus and had enough chances to score an equalizer, but they were unable to take advantage.
Roberto Mancini made eight changes to the Manchester City side that drew 0-0 with Swansea, with one eye on the upcoming FA Cup final against Wigan.
Meanwhile, the away side made four changes, with Craig Dawson awarded his first start in the Premier League this season after returning from a loan spell at Bolton.
The Baggies began brightly as Romelu Lukaku forced a good low save from Joe Hart at his near post and then sent a long-range effort just wide.
Manchester City also started the game positively in what was a very open beginning, with Carlos Tevez playing a neat one-two with Samir Nasri before firing wide.
It took some time for either side to create another chance, as Boaz Myhill spilled a good free kick from Aleksandr Kolarov into the path of Jack Rodwell, whose shot went over the bar.
James Milner was then presented with a glorious chance to give Manchester City the lead, but the England midfielder lashed his close-range shot wide.
The home side then had two more superb openings, as Rodwell could so easily have scored after a corner then Tevez hit the bar soon after.
In the 35th minute, the Citizens were finally rewarded for their persistence as after a good cross from Tevez, Dzeko finished well with a volley to give his side the lead.
The hosts were largely dominant for the remainder of the half, although they could not add to their lead before halftime, although Milner saw a long-range shot hit the post.
West Brom started the second half brightly, and within minutes of the restart striker Markus Rosenberg hit a free kick that rattled the crossbar.
The Baggies continued to impress in the opening stages of the half, with James Morrison hitting a good low shot from range that Hart had to palm away.
However, after initially looking rather lethargic the home side suddenly had a good chance of its own as Rodwell twice saw shots blocked inside the box.
The impressive Milner then had yet another shot on goal as he cut in from the right and had a shot that was gathered comfortably by Myhill.
The Baggies still looked confident of getting on the scoresheet, however, with Rosenberg’s smart shot forcing another good save from Hart.
The away side then had its best chance, as Billy Jones and then Shane Long could not get their efforts on target.
However, despite their continued efforts, the Manchester City defenders held firm and ensured they came away with another home win.