Fulham, in 19th place with 31 points from 37 matches, never looked up for the fight and goals from Peter Odemwingie, Marko Arnautovic and Oussama Assaidi put the home side 3-0 in front.
Kieran Richardson pulled a goal back with 10 minutes to go, but Jon Walters completed the rout two minutes later and, with Sunderland's 1-0 win over Manchester United, defeat has left Fulham four points behind them with one game remaining.
"This is one of the worst days I've ever had," manager Felix Magath told reporters.
"We had no chances from the beginning, we weren't in the game. The team felt too much pressure, we couldn't run, we couldn't pass, we couldn't play.
"Afterwards, nobody said a word. Everyone was down, there were some tears."
Former Bayern Munich coach Magath took over from Rene Meulensteen in February and was handed the task of keeping Fulham in the top flight, but he failed to reignite a disjointed side and won just three of his 11 matches in charge.
"I was very confident that we could avoid relegation and I was convinced before the game that we had a great chance," the 60-year-old German said.
"I spoke with the players before and they seemed to be confident and in the dressing room they were up for the fight.
"But I have to take responsibility and I apologise, not only for today, but for the situation."
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Fulham narrowly avoided relegation under Roy Hodgson in 2008, with the current England manager transforming their fortunes as he led them to the 2010 Europa League final before joining Liverpool later that year.
Mark Hughes took over from Hodgson, spending one season at Craven Cottage before Martin Jol was appointed manager in 2011 and the former Ajax Amsterdam coach guided them to consecutive mid-table finishes.
At the start of this season, owner Mohamed Al Fayed, who bought Fulham in 1997, sold the club to American businessman Shahid Khan and after a poor start Jol was replaced by Meulensteen.
In a bid to climb away from danger, Fulham signed Greek striker Kostas Mitroglou from Olympiakos Piraeus for 12 million pounds ($20.25 million) in January, but injury restricted him to three appearances.
As relegation threatened, Magath replaced Meulensteen and despite being unable to save them he is confident he can lead the club back into the Premier League next season.
"When I joined it was clear we had to change something as the club was at the bottom of the table," he said.
"Things developed in the last few weeks but it wasn't enough. I will talk about the situation with the owner and we will start the comeback."
