Having secured a goalless draw in the semi-final first leg in Germany, the Cottagers were boosted before kick-off by star striker Bobby Zamora being passed fit to start.
Zamora threatened early on as he wriggled clear to get a shot in, but visiting keeper Frank Rost was well-positioned to make a block.
Hamburg then began to enjoy much of the possession, but were unable to forge any meaningful openings despite earning several corners.
However, that changed 22 minutes in when Danny Murphy conceded a free-kick 30 yards out and Mladen Petric's fierce left-foot shot left Mark Schwarzer clutching at thin air as it tore into the net.
The goal left Fulham needing two at least to progress and they took some time to recover from the shock of falling behind.
Seven minutes before the break Jonathan Pitroipa blasted a shot narrowly wide from 20 yards, as Hamburg ended the first half in the ascendancy.
Fulham continued to struggle to make headway after the restart and the tiring Zamora was replaced by Clint Dempsey shortly before the hour mark.
Damien Duff shot narrowly wide of the far post following a slick free-kick soon afterwards and then David Jarolim wasted a good shooting chance at the other end.
The Cottagers then blew the tie wide open as Danny Murphy dinked a superb pass over the defence and Simon Davies brought the ball down superbly, turned past his marker and placed the ball past the advancing Rost.
Fourteen minutes from time the turnaround was complete as a Davies corner from the right dropped eight yards from goal and Zoltan Gera latched onto the loose ball and drove it home to send Craven Cottage wild.
Late on Ruud van Nistelrooy poked a half-chance wide of target and Fulham held on through three added minutes to celebrate wildly at the final whistle.