Captain for the day Samir Nasri got the game off to a lively start when a high clearance from the French midfielder evaded the entire Limerick back line, almost putting out-of-favour Paraguayan striker Roque Santa Cruz through on an open goal.
City broke the deadlock on the half hour when a weaving run from James Milner created space for the England international to roll the ball into the path of Dzeko, who bundled home from close range.
The visitors doubled their lead on 42 minutes when defender Dedryck Boyata received the ball from youngster Denis Suarez and calmly slotted home.
And Dzeko found himself clear of the Limerick defence just a minute later, firing low and hard to put the Barclays Premier League champions 3-0 up.
The second half was played out at a more steady pace, only reigniting on 78 minutes when 21-year-old Italian midfielder Luca Scapuzzi fired home a fourth for City.
Limerick began to press as the game wore on but were unable to find a way through a solid defence.