Dzeko has struggled to make a lasting impact at City during his two years at the club following his big-money move from Wolfsburg in 2011.
Pellegrini told reporters: "Dzeko played well against Newcastle, but not very, very well because for all the things he did he must score.
"Then he will have played very well indeed. He had bad luck. Of course I have no doubt he will score goals. He has scored goals through his whole career.
"I spoke with him before I arrived here and told him that if he will stay here this year he must demonstrate the player that Manchester City bought two years ago."
City face Cardiff City in the Premier League later today.