Not even the return of Martin Petrov after almost six months on the sidelines with a knee injury lifted the home side, who looked lethargic for most of the first half-hour and were short of ideas after the break.
City, who need to score at least twice against Hamburg to turn around their deficit from the first leg, will need a huge improvement on this display if they are to trouble the German side.
It was Fulham who took the initiative in the early stages and they created the first opening when Dunne slipped as he tried to deal with a Simon Davies header. The ball dropped for Bobby Zamora but Shay Given was out quickly to block his shot.
The visitors also had claims for a penalty turned down when Dunne and Andy Johnson tussled in the box, while Dempsey wasted a good opportunity when he slipped with the goal gaping and fired well wide.
City, in contrast, were creating next to nothing yet still took the lead on 28 minutes.
Hodgson wants top-ten finish from Fulham
Ireland picked up the ball in his own half and burst forward before trying his luck from the edge of the area with a shot that clipped John Pantsil and looped into the net.
That goal flattered City but they almost had another before half-time when Valeri Bojinov's shot fizzed inches wide.
Fulham, unsurprisingly, looked frustrated after seeing all their early good work go unrewarded but their perseverance paid off after the break.
In the 50th minute, Zabaleta carelessly lost possession and, although Micah Richards did well to stop Zamora, the ball fell perfectly for Dempsey to beat Given with a powerful low shot.
By now, City were looking vulnerable at the back and paid the price for failing to clear their lines from a cross when Etuhu, whose brother Kelvin was in City's side, lashed a powerful shot past Given from the edge of the area.
And things almost got worse for Hughes' beleaguered men when Brede Hangeland somehow headed over from a Davies free-kick with Given stranded.
The home crowd called for Robinho's introduction and got their wish after 63 minutes.
Hughes admits City lacked 'energy'
But the closest he came to rescuing his side came when he was sent clear by fellow substitute Danny Sturridge but failed to get enough power on his shot to beat Mark Schwarzer.
City, who had won their previous six home games, became increasingly desperate near the end and another mistake allowed Dempsey to make sure of the points after 83 minutes.
Dunne was caught out when he tried to intercept a through-ball on the half-way line and Zamora ran through before slipping the American clear and he fired through Given's legs and into the net.
Schwarzer had to make a smart save from Ched Evans' header but Fulham finished up as comfortable winners.
After waiting until 13 March to record their first away Premier League win of the season, Fulham thoroughly deserved their second victory on the road which keeps their bid for seventh place - and a Uefa Cup spot - alive.
Given, Richards, Onuoha, Dunne, Garrido, Etuhu (Sturridge 64), Zabaleta, Ireland, De Jong, Petrov (Robinho 63), Bojinov (Evans 55). Subs Not Used: Hart, Elano, Fernandes, McGivern.
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