The Barclays Premier League champions were rarely troubled as they eased to a convincing victory at the Etihad Stadium after Zabaleta opened the scoring late in the first half.
Dzeko tapped in for a second just before the hour and the lively Aguero added a third from the penalty spot 16 minutes from time.
The Argentinian did beat Begovic when he got on the end of a Gareth Barry cross but his header came back off the post.
The hosts were finally rewarded for their superiority two minutes before the interval when James Milner's cross was parried by Begovic and the excellent Zabaleta drove home the rebound.
Dzeko was getting closer to troubling his friend Begovic and he accepted a gift from the keeper to double City's lead in the 56th minute. Aguero weaved his way through the area and poked a shot at Begovic which, unsighted by a defender, he could only parry and Dzeko pounced inside the goal area.
Steven Nzonzi was then the guilty party as City all but wrapped up the game from the penalty spot in the 74th minute. There may have been debate over whether David Silva was in the area when touched by the former Blackburn midfielder but there was no doubting the contact and Aguero struck from 12 yards.
It proved Aguero's last meaningful contribution, however, as he pulled up soon after with what seemed a hamstring injury and he was replaced by compatriot Carlos Tevez.
The game started to peter out but Stoke had a late flurry as Glenn Whelan, with the possible aid of a deflection, brought a fine save out of Joe Hart with a long-range low shot, which was tipped onto the post.