The Bosnian opened the scoring against the Czech champion, Yaya Toure added the second, while Sergio Aguero put the seal on a fine away win for Manuel Pellegrini's men.
City scored three goals in 10 second-half minutes at the Doosan Arena to consign the Czech champion to defeat in the Group D encounter.
Striker Dzeko opened the scoring and Yaya Toure added a stunning second before Sergio Aguero, who also struck a post, capped a fine individual display with a third goal after 58 minutes.
Big-spending City has failed to progress from the group stage in the last two seasons but will have high hopes of reaching the knockout stages after such a promising opening result.
"It was the perfect start," Dzeko told Sky Sports. "The last two years we didn't do that. We know that every team in the Champions League is strong and we are very happy to go home with three points."
City did not have things all its own way as Plzen, which came through three qualifying rounds to reach this stage, gave a good account of itself in the first half.
City dominated after the break, however, and ultimately ran out a comfortable winner.
"As you can see, after the first goal, we scored two more and finished the game - but in first half it was a little bit different. We had some chances to score then after I scored it was easier," Dzeko added.
The former Wolfsburg man is enjoying life under new boss Manuel Pellegrini, who left close-season signings Alvaro Negredo and Stevan Jovetic on the bench.
When asked if he was pleased the new manager has put his faith in him, Dzeko continued: "It is very important. We have a very good squad with some new players and they've brought a lot of quality. There are a lot of games until the end of the season and everyone will get a chance to play."
Meanwhile, Toure added another spectacular strike to his collection when he found the top corner from 25 yards. The Ivory Coast midfielder was similarly delighted to make such a positive start to the campaign.
"We had the desire, we wanted to win the game," he said. "It is always very hard in the Champions League and they played well.
"The last two years were disappointing for fans and the club. The target is to go to second round and as far as possible after that but we will see."
City's next Group D outing is a mouth-watering match-up with reigning champion Bayern Munich at the Etihad on Oct. 2.