City will travel to Leicester City in the quarter-finals of the competition after securing a win at St James' Park after extra time on Wednesday night.
Dzeko played a pivotal role, setting up the opener for Alvaro Negredo, before scoring the second himself.
And the striker thinks it was an important night for players on the fringes of City's first team.
He said: "Tonight was a chance for those of us who maybe haven't played in the last few games to show the manager we deserve to play.
"We have to show it on the pitch and I think we did that tonight."
Dzeko was also full of praise for goalkeeper Costel Pantilimon, who won man of the match and is now putting Joe Hart under pressure for the number one shirt.
"He was great," Dzeko said. "He's working hard in training and that is now the second game in the cup where he hasn't conceded a goal, so he's improving."
The cup may be low on City's priority list but Dzeko insists they still want to win it.
He added: "There's still three more games to the final but we definitely want to go there - that's our target."