Neal Bishop's header on the hour threatened to add City's name to the Sunderland scalp County collected in the previous round.
Shocks in the FA Cup are almost as old as County themselves and, for 20 minutes, the League One outfit thought they were going to claim another.
But Dzeko had other ideas and drilled home Micah Richards' low cross 10 minutes from time to break his own City duck, and rescue the Eastlands outfit who must play again in three weeks' time for the right to reach the last 16.
Having struggled to dispose of Sven-Goran Eriksson's Leicester in the last round, Roberto Mancini would have expected to be similarly troubled by a team managed by Paul Ince.
After all, their shared history includes a red card for Mancini for a tackle on Ince during a particularly ill-tempered Serie A spat between Sampdoria and Inter Milan.
And the world's oldest Football League club did not disappoint, matching City blow for blow on a pudding of a pitch, churned up badly by the two rugby matches that had been played on it over the past week.
Certainly Jon Harley's tackle on James Milner was straight out of the 15-a-side code's textbook, although as referee Chris Foy elected to show Mike Edwards the yellow card, perhaps he did not get a good view.
By that point City should have been ahead from the only decent opportunity of the opening 45 minutes.
Micah Richards burst down the right flank and sent over a low cross to the near post.
Yaya Toure got his not-inconsiderable frame to the ball first and prodded it goalwards.
But Stuart Nelson was alive to the danger and turned it away with an excellent feet-first save.
Having opted to rest Carlos Tevez, Mancini handed £27million new-boy Dzeko his FA Cup debut.
The Bosnian was still searching for his first City goal and had a couple of glimpses before breaking his duck.
One shot from the angle of the area soared disappointingly over, then Krystian Pryce needed to make an excellent tackle to prevent the Bosnian trying his luck from very close range.
Another opportunity came his way immediately after the break too when Richards went on another overlapping run. For the money he cost, the Blues might have expected more than a scooped shot that did not threaten the home goal.
It was a costly miss too as Lee Hughes made his muscular presence felt in an aerial duel with Joleon Lescott which he was convinced should have brought about a penalty.
Referee Chris Foy only gave a corner, which as it turned out, was just perfect for the League One outfit, whose next opponents are Walsall on Wednesday.
It was that type of opposition Bishop was dreaming of playing against when he fulfilled his ambition to play in the Football League with Barnet, aged 26.
Those thoughts might well have gone flashing through his mind as he was submerged by delirious team-mates after rising to meet Gow's near-post corner and glancing the ball beyond Hart.
The introductions of David Silva and Aleksandar Kolarov were designed to generate some attacking impetus, of which there was precious little from City beyond a dipping Gareth Barry effort that Nelson did well to tip over.
But those who question the resolve within Mancini's squad have clearly got it wrong.
With time ticking away, a touch of quality from Silva released Richards once more.
As he had done before, the full-back crossed low and hard. This time, from six yards, Dzeko did not miss.
The hosts did not just accept their fate. Edwards galloped in at the far post but failed to get a touch to Harley's free-kick as it flew an inch wide.
Hughes went close to grabbing a last-minute winner too, and although City were on top at the end, a draw was the least County deserved for their efforts.