The Cottagers had gone ahead early on when Zoltan Gera's low shot crept in.
Dejan Stefanovic could only deflect a Zoltan Gera cross and Horsfield struck at the far post.
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Zoltan Gera slotted Fulham's third before Marco Streller nodded home a late corner to ensure a tense finish.
Zamora cushioned Paul Konchesky's cross into the path of Gera, but the Hungarian's half-volley bounced over the bar.
Ruud van Nistelrooy had an early half-chance before the superb Zoltan Gera saw a shot deflected narrowly wide.
Gera found the time in the area to line up a shot which was superbly blocked by Roger Johnson.
But Danny Murphy's penalty drew Fulham level early in the second half before Zoltan Gera put the visitors ahead.
The West Brom boss also revealed he wants to start talks with out-of-contract pair Zoltan Gera and Kevin Phillips.
Robson made a surprising double substitution on the hour, replacing Zoltan Gera and Earnshaw with Geoff Horsfield and Rob Hulse.
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Fulham lost Bobby Zamora to injury but his team-mates coped well and Gera lost his balance when through on goal.
Midfielder Danny Murphy started to move further forward and Gera and Carlos Salcido attempted shots, though both failed to trouble Howard.
Do-Heon Kim headed them ahead from Zoltan Gera's cross and Chris Brunt sealed the win by firing home a superb free-kick from 20 yards.
Fulham almost made the Blues pay late on as both Dempsey and substitute Zoltan Gera drew good saves from Petr Cech with well-struck long-range shots.
Zoltan Gera took his place on the left, with Simon Davies making his first start of the season following an ankle injury at Erik Nevland's expense.
And Fulham eventually buckled when John Arne Riise's shot took deflections off Gera and his brother Bjorn on the way past wrong-footed Cottagers keeper Mark Schwarzer.
Zoltan Gera replaced the stricken Zamora and although John Pantsil rifled a shot over the bar, Fulham offered little threat for the rest of the half.
But it was Fulham whose play carried the greater danger and Kamara, Dempsey and Gera were combining well in attack, with an assured looking defence behind them.
Joris Mathijsen had to be at full stretch to cut out a teasing cross from Gera, who then broke free in the box but lost his footing.
An unmarked Kamara headed wide at the far post before Dempsey angled a shot wide after being set up by Gera and Nevland's red card then proved decisive.
But the game turned Albion's way eight minutes after the break when half-time substitute Kim headed in Gera's cross from close range to give the visitors the lead.
"She was a GREAT and comforting role model for single mothers, as I was one of them at that same time, " wrote a CNN.com commenter with the handle Gera Merix.
Zamora's replacement Clint Dempsey, back after three games on the sidelines, looked particularly sharp and had a penalty appeal turned down before linking well with Gera to carve out two openings.
Again Zamora was central, flicking cleverly to Davies on the left and from the midfielder's first-time cross Gera was on hand to side-foot into the roof of the net from six yards.
Hodgson brought on Diomansy Kamara to replace Nevland on the hour and Fulham, at last, showed glimpses of creativity as Zoltan Gera, Dickson Etuhu and substitute Seol Ki-Hyeon had chances to score.
Brede Hangeland nodded a corner wide, Zamora shot straight at keeper Povilas Valincius and Zoltan Gera flashed wide of the near post as the Cottagers failed to turn their increasing dominance into goals.
Suddenly the Cottagers were rampant and they went close to levelling soon after, when Alex Manninger clawed away a Brede Hangeland header, then threatened again with shots from Zoltan Gera and Bobby Zamora.
Mladen Petric's vicious free kick had given Hamburg a half-time lead in the second leg at Craven Cottage, but Simon Davies and Zoltan Gera scored the two goals required to take Fulham though.
Paul Konchesky arrowed a pass upfield and Zamora laid it into the path of Gera, whose shot deflected off Jerome Boateng and squirmed just wide of Frank Rost's upright with the goalkeeper wrong-footed.
Gera then volleyed over on the turn and Dembele forced Hahnemann into a low save before Davies appeared to be brought down by the American as he tried to take the ball round him.
Danny Murphy split two defenders with a pass to the edge of the box to Zamora and after his cute lay-off, Gera lashed under the body of Pyatov from an angle on the left.
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