City made a bright start to the game in front of a noisy and enthusiastic Eastlands crowd and forced their visitors to defend in depth.
However, Hamburg struck a killer blow with 12 minutes on the clock as Jonathan Pitroipa made a dash down the right and his cross was deflected into the path of Jose Paolo Guerrero, whose low shot beat Shay Given from ten yards out.
City were handed something of a lifeline within a matter of minutes as Piotr Trochowski was adjudged to have handled Elano's attempted shot.
The Brazilian stepped forward to take the spot-kick himself and calmly beat Frank Rost to his right to make the score 1-1 on the evening and 4-2 on aggregate.
Trochowski almost rectified his error midway through the first half as he tested Given from distance and ten minutes before the break, Felipe Caicedo wasted a chance from close range at the other end.
Just before the interval City came within inches of taking the lead on the night, as Elano smacked a 25-yard free-kick against the crossbar with Rost rooted to the spot.
Five minutes into the second half City did grab a second goal as Stephen Ireland's pass fell to Caicedo, who sidestepped a defender before burying a low, left-foot shot under Rost from 15 yards.
Soon after Elano saw his 25-yard free-kick deflected against an upright, before Ivica Olic lifted a chance over the top at the other end.
On the hour, City thought that they had levelled the aggregate scores when Caicedo raced through to finish, but an offside flag cut short the celebrations of the Ecuador striker.
The game was now being played mainly in the Hamburg half, with Elano and Robinho causing the visitors' defence plenty of problems.
However, 15 minutes from time the home side were reduced to ten men as Richard Dunne received a second yellow card for a foul on sub Mladen Petric and was ordered off.
With six minutes remaining Olic should have tied things up for Hamburg, but his close-range shot was superbly deflected away for a corner by the alert Given at his near post.
Micah Richards skied a good late chance to take the tie into extra-time, but Hamburg held out to reach the last four and end English interest in the competition.