Second-place City is now assured of finishing in the top three, which means a direct spot into the group stage of Europe’s premier club competition.
City rested a number of its top players ahead of Saturday’s FA Cup final against Wigan, but Dzeko guided home a volley in the 35th minute for his third goal against West Brom this season.
?Both sides spurned numerous other chances with City hitting the woodwork twice through Carlos Tevez and James Milner in the first half and Marcus Rosenberg doing likewise for West Brom after the break.
Romelu Lukaku had three good chances inside the opening six minutes but failed to trouble Joe Hart, shooting and heading at the keeper and dragging another effort wide.
Tevez had City’s first attempt after linking well with Samir Nasri and cutting inside but his shot was well off-target. Lukaku teed up Rosenberg at the other end but he too fired the wrong side of the post.
City then enjoyed a good spell as Tevez shot narrowly over from the edge of the box and James Milner slammed a good chance from a dangerous Nasri cross.
Jack Rodwell was then inches away from turning Kolarov’s corner towards goal and Tevez rattled the crossbar with a powerful drive from the edge of the box.
The opening goal came moments later as Tevez broke away down the right and clipped back a delicate cross for Dzeko. The Bosnian, guilty of a glaring miss at Swansea, watched carefully and guided a well-weighted volley into the net.
Mancini told Sky Sports: "I thought that it was a difficult game. We changed a lot of players, we took some risks, but in the end it was important to win this game.
"After 10 minutes, we started to play very well. We had five incredible chances to score, but we were missing. For that reason, we suffered more in the second half. They are a good team. They have had a fantastic season, they play good football. We knew that it would be a difficult game."
Asked if he has now settled on a side for a trip to Wembley this weekend, Mancini added: "Maybe. We have four days. It is important after this game to recover our strength. We didn’t take a risk with some players. I think so."