David Silva grabbed both goals, the first inside two minutes and the second a sublime chip, in a 2-0 victory over Fulham at the Etihad Stadium, their fourth win in succession.
City have recovered their poise since losing to United and Sunderland in December and are determined not to give up their crown without a fight.
Dzeko told the club's official website: "We have to just keep doing what we're doing.
"I think Manchester United will drop points sooner or later but we just need to keep winning and keep the pressure on them."
City were the dominant team and although Fulham frustrated them at times in the 67 minutes between Silva's strikes, they could have won by a greater margin.
Silva was inches away from a hat-trick when he rolled a shot past the far post just moments after making it 2-0 and late substitute Sergio Aguero had two good chances on his return from injury. Dzeko also had opportunities, not least when he fired over from close range in injury time.
Dzeko, City's top scorer with 11 goals in all competitions this season, said: "David took his goals really well and he is disappointed he didn't maybe get a third but I'm really pleased for him and that the team won.
"We are a bit disappointed we didn't score more goals after taking the lead so early in the game and I think if we'd scored again in the first half we'd have gone on and won by more but sometimes it doesn't pan out like that.
"We got the second at a good time so we have to be happy. I'm disappointed I didn't score, but I think I played okay. As long as we keep winning, I'm happy."