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.
BBC: SPORT | Football | Premiership | West Brom 2-0 Portsmouth
Zoltan Gera slotted Fulham's third before Marco Streller nodded home a late corner to ensure a tense finish.
Ruud van Nistelrooy had an early half-chance before the superb Zoltan Gera saw a shot deflected narrowly wide.
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.
BBC: SPORT | Football | Premiership | West Brom 1-2 Middlesbrough
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.
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.
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.
He broke the deadlock in the 59th minute, collecting an infield pass from Zoltan Gera before manoeuvring himself into a shooting position on the edge of the box and curling a superb shot past keeper Diego Benaglio.
The goal, which saw him collect Zoltan Gera's pass, turn Jan Simunek and slot home, was Zamora's sixth in Europe this year and his 19th in total in what has been an excellent season for the striker.
Zoltan Gera slid a pass into the Roma box for Clint Dempsey and he cut the ball back for Kamara, who was tripped by Marco Andreolli - the scorer of Roma's equaliser in the reverse fixture two weeks ago.
Fulham had less of the ball than Everton boss David Moyes's team and it took well over half an hour before Zoltan Gera produced their first shot, when Dickson Etuhu marauded into the box only to place his header straight at Howard.
He not only rejuvenated the likes of Danny Murphy and Simon Davies, but identified unheralded talents such as Brede Hangeland and experienced players who had fallen out of favour at their previous clubs - Mark Schwarzer, Zoltan Gera, Bobby Zamora and Damien Duff to name but four.
Hodgson's carefully crafted team of budget signings (Zoltan Gera, a free transfer from West Brom), obscure European finds (Brede Hangeland from FC Copenhagen) and rejuvenated 'Big Four' cast-offs (Danny Murphy, ex-Liverpool, and former Chelsea winger Damien Duff) have negotiated 14 games on their way to their maiden European final.
The in-form striker, who passed a fitness test before the game on an Achilles injury, collected a pass inside the area from fellow forward Zoltan Gera and stylishly set up a cool, side-foot finish past home keeper Diego Benaglio for his sixth Europa League goal of the season and 19th in total.
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