The world's richest club took their time to break the deadlock but were solidly on top from the middle of the first half and could have gone ahead when Silva and Gareth Barry rattled the bar while Yaya Toure blasted wide.
Bosnian striker Dzeko eased any Eastlands nerves when he tucked the ball away in the 57th minute after Swansea's busy Dutch keeper Michel Vorm, also making his league debut, parried Adam Johnson's shot following a superb driving run by Silva.
Aguero, having had no real practice in a City shirt after finishing the Copa America last month, came on just after the goal and almost netted with his first touch as he made one of the most memorable debut cameos English football has seen in recent years.
"There is a great bond in this group, I'm feeling the love," City keeper Joe Hart said.
Manchester City's Edin Dzeko celebrates with teammate Yaya Toure after scoring the opening goal
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