Everton earn a point in a 1-1 draw against Manchester City at the Etihad Stadium.
Marouane Fellaini gave the Blues a deserved lead just after the half-hour mark but a Carlos Tevez penalty on the stroke of half-time earned the champions a share of the spoils.
Moyes named an unchanged team to the side that drew against Arsenal in mid-week, with Leighton Baines â€“ a major doubt before the game â€“ passed fit to start and Toffees talisman Fellaini lining up behind Nikica Jelavic.
Mancini rotated his expensively assembled squad, making four changes to the team that beat Wigan 2-0 on Wednesday night. Joleon Lescott came back into the side to face his former club, while Samir Nasri, Tevez and Dzeko also started.
Everton made the brighter start, showing more tempo than the home side in the early exchanges, with Cityâ€™s technicians struggling to affect the game before briefly bursting into life mid-way through the first half; David Silvaâ€™s cross was deflected inches wide by Sylvan Distin before Nasri had a goal-bound shot blocked by the head of Dzeko.
But the Toffees were rewarded for their early endeavour in the 33rd minute. Baines, afforded too much space from deep, had time to swing in a brilliant far-post cross which was met with a powerful header by Fellaini. Joe Hart did brilliantly to stop the initial shot, but palmed the ball more up than out and the Belgian was able to tuck in the follow-up.
City mounted an energetic response, with Tevez flicking a header towards the back post that was well saved by Tim Howard before Nasriâ€™s cross was chested down by Gareth Barry for Dzeko to blast towards goal.
From the resulting corner, Dzeko was unnecessarily manhandled by Fellaini, with referee Lee Probert pointing to the spot. Tevez stepped up, waited for Howard to dive to the right before passing the ball down the middle of the goal. Fellaini almost made amends before the half-time whistle, forcing Hart into a good low save after outjumping Vincent Kompany.
There was more energy and control to Cityâ€™s play after the break, with the tenacious Tevez leading by example; but the Toffees, in typical Moyes fashion, defended resolutely and the Citizens struggled to carve out any clear-cut chances.
Mancini tried to rally his side with the introductions of Sergio Aguero and Mario Balotelli, but it was to little avail and it was the visitors who came closest to scoring. Kompany cynically ended a swift Toffees counter, and Hart was almost embarrassed by Jelavicâ€™s resulting free-kick.