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Tevez had a chance to get a second in the 18th minute when he rose highest to meet Aleksander Kolarov's cross, but his header went high over the bar.
Dzeko was the next guilty party to miss a good headed chance, turning his effort wide from eight yards out having met Nasri's corner.
City did double their lead in the 28th minute when Tevez drilled a ball across goal which Kolarov met to bury into the bottom corner.
It was 3-0 just two minutes later and again Tevez was involved. The Argentine was found by Silva in the box before turning and firing his effort past Steele from six yards out.
Tevez almost bagged a first-half hat-trick with seven minutes left, but his shot from the edge of the area was tipped over the top by Steele.