Dzeko then scored a crucial third goal for the visitors at the start of the second half before Romelu Lukau pulled a goal back for Everton on 65 minutes.
Read Sports Mole's minute-by-minute report below to find out how the action unfolded at Goodison Park.
On target: 4
On target: 6
Was the result fair?
While there was little to separate the teams most of the match, Man City's killer instinct in front of goal saw them come away with a well-earned three points. They may have failed to do that in the past, but this team seems to be more focused on what it takes to win.
The Toffees started solidly and took a surprise lead through Barkley's stunning effort, but they found it hard to break their opponents down after that. There was plenty of dynamism and commitment from the hosts, particularly from Barkley and Steven Naismith, though it was in the final third where they continuously let themselves down. Despite repeatedly trying to get in behind the Man City back four, Roberto Martinez's side were shown to be lacking in ideas by the end of the evening. It has failed to happen for Everton in recent weeks, and this was another example of that.
Man City's performance
This was a professional, clinical display from City as they took a significant step towards a second Premier League crown in three seasons. They were their usual selves in attack as Dzeko, Aguero, Yaya Toure and Samir Nasri caused Everton repeated problems, but plenty of credit should go to captain Vincent Kompany and his fellow defenders for their resolute attitude at the back. Not only did they successfully soak up most of the pressure that Everton put on them, but they also stifled the hosts from creating any meaningful opportunities. This was a huge test for Manuel Pellegrini's side, and they came through it with flying colours. You would not bet against them lifting that trophy next weekend now.
Sports Mole's man of the match
Ross Barkley: The Everton youngster gave a supreme performance in midfield, running from deep in his own inimitable way while causing Man City more problems than any other player in a blue shirt. Barkley also lit up the match early on when he scored with an eye-catching effort from outside of the area. What a prospect he is.
Dzeko made a bit of a fool of himself towards the end of the match when he stayed down on the ground with what appeared to be a facial injury, although it was clear that the Bosnian was making a meal of it. Lee Probert waited an eternity for him to get up before booking him for an earlier foul.
Lee Probert had a largely solid afternoon, but there were one or two decisions that were questionable. He could easily have sent off Fernandinho for a elbowing Naismith in the throat late on, while Lukaku could quite possibly have been offside for his goal.
Everton: The Toffees wrap up their Premier League campaign next Sunday when they make the trip to Hull City.
Man City: City play their all-important game in hand on Wednesday as Aston Villa travel to the Etihad Stadium.