Premier League: Man City 1 Crystal Palace 0
The Bosnia-Herzegovina international striker converted Jesus Navas' cross midway through the second half on Saturday to end Palace's stubborn resistance and make it 10 wins out of 10 at Etihad Stadium in the league for Manuel Pellegrini's side.
Victory puts City two points clear at the top of the table, with title rivals Arsenal, Chelsea and Liverpool all in action on Sunday.
Palace had chances of their own in the second half, with Joe Hart tested on three occasions, while Joel Ward missed a golden chance to equalise a minute after Dzeko's strike, heading off target when left all alone at a corner.
Pellegrini made six changes to the side that beat Liverpool on Boxing Day, with Yaya Toure and Alvaro Negredo among those to drop to the bench, while Palace were unchanged from their win at Aston Villa.
The hosts dominated the early stages as Dzeko, David Silva and James Milner all fired half chances off target in the opening 25 minutes.
The action came to a halt on the half hour as Hart and Cameron Jerome both received treatment following a collision. Jerome left the field to be replaced by Marouane Chamakh, while Hart remained in goal, despite a nasty cut below his left eye.
City continued to press after the stoppage in play, Silva blazing an effort high and wide of the target after a deft touch on the byline before Fernandinho thumped a volley over the bar a minute before the break.
The Brazilian went even closer in first-half stoppage time, heading Milner's cross goalwards only to see Julian Speroni leap to tip the ball behind.
Hart was stretched for the first time in the 50th minute, diving low to his left to keep out Jason Puncheon's arrowing 25-yard drive before Chamakh headed the resulting corner over from six yards.
The England international goalkeeper had to be alert again 14 minutes later when he tipped Mile Jedinak's rasping drive over the bar.
That save proved crucial as two minutes later the deadlock was finally broken, Dzeko sweeping home Navas' cutback from 12 yards as City made it 60 consecutive home league games in which they have found the net.
Palace almost responded immediately, Hart diving full length to his left and just getting enough on the ball to tip Ward's volley around the post before the same player wasted a glorious chance from the resulting corner, heading wide when left unmarked six yards out.
It was a costly miss as City held on for the win to extend Tony Pulis' miserable record at the Etihad - the Welshman has now lost all six Premier League visits to the ground without scoring.