Goals from David Silva and Yaya Toure in either half gave City the win they needed but Napoli's 2-0 success at Villarreal helped the Italian club progress alongside Group A winners Bayern.
Despite notching what is often seen as the magic 10 points that usually takes teams through, City finished third in what manager Roberto Mancini called the "worst group" and will now go into the second tier Europa League.
"I think that usually with 10 points all the teams go through always, 99 percent," Mancini told a news conference.
"But in this group it was not enough, Napoli got one point more than us and they deserved to go through. We made some mistakes but normally with 10 points you can go through."
They drew 1-1 at home to the Italians in their group opener and lost 2-1 away from home despite dominating for long spells in both games, with Mancini saying the second game had "probably" been the one that had cost them qualification.
The hosts had the ball in the back of the net in the 15th minute when Silva's free-kick was headed in by Joleon Lescott but it was disallowed for a foul by the defender on Bayern keeper Joerg Butt.
They started the second half as they ended the first, doubling the score in the 52nd minute when Aguero played the ball through to Dzeko who had spotted Toure's perfectly timed run and the charging Ivorian slotted it past Butt.
For some, the blow was softened slightly by the fact neighbours and rivals Manchester United had suffered the same fate after losing 2-1 at FC Basel while others will have noted Mancini's vow to take the other European competition seriously.
"For Manchester City it's an important trophy because we need to win more trophies," said the manager, who guided City to thei
