City won the second leg of their tie against Hamburg 2-1 at Eastlands but lost 4-3 on aggregate.
Elano and Felipe Caicedo scored but Elano also hit the woodwork twice and Caicedo missed a glorious chance from a few yards out as City, who also had Republic of Ireland defender Richard Dunne sent off for picking up two bookable offences, came up just short.
Ireland said: “We had a lot of chances and we didn’t take them, we should have been more clinical.
“I think Caicedo had a few clear-cut chances and didn’t finish them, the same with Micah Richards.”
City were outplayed in the first leg but turned things around impressively at home and Ireland has called on the Blues to ensure they secure seventh spot in the Barclays Premier League and a return to Europe.
“We have to have that spirit every week,” the Irishman added. “We have to pick ourselves up. We’ve got to get seventh if we can and not let this destroy our season.”
City boss Mark Hughes felt his side did not get the slice of luck they needed in order to at least force extra-time. He said: “We’re proud of the performance. It was a big night for us and we’ve produced a real performance of desire and commitment.
“The crowd sensed something really special was happening but we just needed that break at the right time.
“We kept on probing and kept trying to do the right things, putting pressure on their back four and they couldn’t handle it. They’re a good side and we made them look quite ordinary at times.
“That’s the standard and we’ve got to do it more often.”
Hughes echoed Ireland’s belief that City can claim seventh spot and a place in next season’s Europa League.
“On the back of that performance, I think so,” he added. “It was important we made a statement and it was there for everyone (to see).”
After being 3-1 down from the first leg the last thing Hughes wanted was for his side to concede an early goal. But that’s exactly what happened and Hamburg extended their lead after only 11 minutes.
City failed to deal with a routine cross from Jonathan Pitroipa and Paolo Guerrero took full advantage to score from close range.
However they battled back and goals from Elano – with a penalty – and Caicedo gave them hope.
But City were left cursing their luck as Elano twice had superb free-kicks come back off the woodwork.
Elano put them level on the night after 16 minutes when Piotr Trochowski was adjudged to have handled his shot. The Brazilian stepped up and converted the spot-kick past goalkeeper Frank Rost to hand City a lifeline.
And after the woodwork had denied Elano City went ahead 2-1 on the night in the 50th minute when Caicedo used his strength to carve out an opening after a clever through ball from Ireland. Caicedo showed good composure to rifle a shot beyond Rost and narrow the margin overall to one goal.
City nearly drew level on aggregate in the 53rd minute only for Elano’s free-kick to rattle the post before Caicedo hoisted a shot over the bar from six yards. Micah Richards and Sturridge both came close late on but the tie ended in frustration for City.
MAN CITY: Given, Richards, Onuoha, Dunne, Bridge, Zabaleta (Fernandes 77), Kompany, Ireland, Elano (Sturridge 84), Robinho, Caicedo.
HAMBURG: Rost, Boateng, Gravgaard, Mathijsen, Jansen, Pitroipa, Jarolim, Aogo, Trochowski (Petric 73), Guerrero, Olic.
Ref: Nicola Rizzoli (Italy).