Dzeko dropped down the pecking order under former manager Roberto Mancini, but started the club’s resounding 4-0 victory over Newcastle on Monday night.
The Bosnia international linked well with Sergio Aguero and David Silva but Pellegrini insists he must strive to score more goals if he is to retain his place over the length of a full campaign.
| MANCHESTER CITY LATEST
|25/1||Edin Dzeko is 25/1 with bet365 to be Premier League top scorer this season
"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.”
Dzeko’s future at the club looked in doubt after the conclusion to last season and speculation suggested he would be one of the first to depart the Etihad Stadium.
Yet Pellegrini, with the help of members of his City squad, have restored the striker’s confidence.
He added: "Dzeko and a lot of players interest a lot of clubs. It’s very important for him to know that all of us trust him, not only me but also the other players, the club, everybody.
"I don’t expect a certain number of goals. The first thing is that the team wins. If the team wins, who scores is not the most important thing.
"Normally the striker will always be scoring, but in my team, normally all the players score goals. That’s very important.
"I don’t want to talk about last season. It’s not my duty. I know what I must do. My duty is to talk with all the players, not just one or two.
"They must know what I want of them and they must know that they are here in this squad because I trust them."