Dzeko scored with just 44 seconds on the clock - the quickest away goal at Old Trafford in Premier League history - after a last-ditch tackle thwarted David Silva and Samir Nasri's shot came back off the post.
The Bosnian striker then ensured City would not pay for failing to press home a dominant first-half display when he caught United napping to volley home a Nasri corner shortly after the break.
Yaya Toure applied gloss a to a fifth win in six league derbies for the blue half of Manchester, and the 3-0 triumph means Manuel Pellegrini's team are now three points behind table-toppers Chelsea with two games in hand.
Dzeko told Sky Sports: "It was an amazing game from us from the first to the last minute. We were amazing.
"We just wanted to start from the first minute to press them high and score an early goal which we did.
"Then it was all about us. We scored another two in the second half so we have to be happy."
And Dzeko, who became the fourth City player to bring up 20 goals this term, believes the team holding their collective nerve in a high-pressure fixture could prove pivotal in the final reckoning.
"Before this game we had (played) three games less than Chelsea and this game was very important for us to keep (up) there," he added.
"Now we are only three points behind and still two games less, so this game can be crucial at the end of the season."
