The fiery South American joined Dimitar Berbatov at the head of the Premier League goalscoring charts with a second-half double to take his tally to 14, accompanying Kolo and Yaya Toure on the scoresheet as the brothers achieved the rare feat of finding the net in the same top-flight game.
It must have been quite an eye-opener for Dzeko, who produced an encouraging performance on his debut and appears willing to do the hard work in his partnership with Tevez in addition to the glamourous stuff.
The Bosnian was also given an instant lesson in the difficulty of securing points in England as City, having been totally outplayed for the first half-hour, then needed to resist a late Wolves surge after the plucky Black Country outfit had pulled themselves back into a contest that had already been turned on its head after their initial heroics.
Having flexed their mighty financial muscle to make Dzeko the sixth most expensive player the Premier League has ever known, City cannot have expected to be rolled over quite so easily as they were in that opening 30 minutes.
It should not have come as a total surprise.
After all, Wolves beat City at Molineux earlier this season and accounted for both Liverpool and Chelsea during the Christmas period, so they clearly cannot be taken for granted.
Still, the one-sided manner of the contest was a shock, the only consolation for Roberto Mancini coming from the knowledge his side remained in the game.
They were already wobbling when Kolo Toure blasted a clearance into Vincent Kompany, the rebound dropping kindly for Nenad Milijas, who finished at the second attempt despite Joe Hart’s initial heroics.
Hart denied Milijas shortly afterwards when the Serbian powered a header towards goal, although there was nothing he could do to prevent Steven Fletcher picking out an unmarked Matt Jarvis at the far post.
Jarvis must have thought he had scored when he pulled the trigger on a shot from 12 yards. Instead, Aleksandar Kolarov flung himself in the way to preserve a fragile state of affairs.
The value of Kolarov’s block was not immediately apparent as it took City some time to assert themselves.
But once Adam Johnson had forced Wayne Hennessey into an excellent feet-first save, Wolves began to assume the air of a side who could not wait for the interval to come.
Unfortunately for them, the clock did not speed round quickly enough.
Tevez was only just wide with a fizzing shot, then, after Dzeko had been denied by an excellent Ronald Zubar challenge, Kompany got his head to Johnson’s corner and flicked it towards the far post.
Kolo Toure controlled, then went for goal, ignoring the crowd of bodies in his way. His decision proved correct as the shot beat Hennessey, bounced through David Jones’ legs and eventually over the line.
It was the Ivorian’s first goal of the season. More importantly, it created the platform for normal service to be resumed after half-time.
More than anyone, Tevez had been inconvenienced by Dzeko’s arrival, shunted from his favoured central berth to a role wide on the left at kick-off.
The Argentinian is not one for keeping quiet, though.
And, as spectators were still taking their seats, he picked up possession on the angle of the area and exploded into life, bursting through the passive presence of Christophe Berra and Stephen Hunt, then skipping inside Richard Stearman.
The hard work done, the rest was easy as Tevez calmly slipped City in front.
Mancini’s men were not done either.
Their third owed much to a neat one-two between Dzeko and Tevez just inside the Wolves half.
The Bosnian then sent Yaya Toure clean through with a perfect through ball, the midfielder following his brother onto the scoresheet with clinical efficiency.
Dzeko came close to a debut goal when his deflected shot flew narrowly wide, but Tevez was in no mood to offer reprieves and when countryman Zabaleta crossed from the right, the Argentina star’s header bounced in via the underside of the crossbar.
Amazingly, within seconds Wolves had pulled one back, Doyle getting to his feet to score from the spot after he had been bundled over by substitute Joleon Lescott.
Zubar added another to set up a frantic ending, which would have been sickening for City had Geoffrey Mujangi Bia’s stoppage-time shot deflected into the net off Kompany rather than just wide.