With finishing like this, Fulham will meet an unhappy end. Felix Magath's side produced their most vibrant performance of the season but squandered chances to convert fine play into goals and ultimately wound up with yet another defeat. A fixture in the top flight since the start of the millennium, Fulham now have only six matches in which to hoist themselves off the bottom of the Premier League.

Everton, meanwhile, look forward to reaching rare heights. Their fifth win in a row did not seem probable in a first half in which they were outplayed but key contributions from their substitutes, on top of a large slice of luck in the form of an own goal by David Stockdale, swung the match in their favour. Roberto Martínez's team are now only four points behind fourth-placed Arsenal with a match in hand. Arsenal go to Goodison next Sunday. "It is a phenomenal challenge but one we embrace," says Martínez.

That showdown would not have got such hot billing if Fulham had fired in front of goal. After 31 matches of their campaign Fulham had long since aborted Plans A and B so for this, their 32nd game, Magath resorted to what must be about Plan X. He gave a first senior start to Moussa Dembélé, a 17-year-old French striker who, it was hoped, would provide the cutting edge that the seven other strikers used by them this term had failed to deliver. The youngster played well but could not give his team any more penetration.

Fulham could have taken the lead in the third minute but Lewis Holtby slashed wide from 15 yards after collecting a cross from John Arne Riise. That was the first, but not the worst, of many spurned opportunities. The home side were furious to be denied a penalty in the 13th minute when Sylvain Distin appeared to handle the ball as he tried to beat Dembélé to a Holtby cross from the left.

The referee, Anthony Taylor, rejected the appeals so Fulham attempted to dipsense justice themselves, but Brende Hangeland's header from a corner dropped inches wide. Kieran Richardson then missed a glorious chance after being teed up by Dembélé, the midfielder's weak low shot from 10 yards allowing Tim Howard to save with his feet.

Everton threatened only through Gerard Deulofeu, who pestered Richardson and Riise, in particular, with his wriggly runs down the left. Deulofeu nearly finished off a sweeping Everton counterattack in the 24th minute but his low drive was turned away by Stockdale. The ever-inventive Holtby conjured a wonderful opening for the hosts on the stroke of half-time, dinking a pass Dembélé at the back post, but the teenager's header lacked enough power to beat Howard.

The England manager, Roy Hodgson, was in the stands to watch Ross Barkley and Leighton Baines – and, perhaps, to give a poignant reminder to the home fans of how quickly Fulham's fortunes have fallen; it was only four years ago that Hodgson led them to the Europa League final.

Barkley did little to enhance his chances of being selected for the World Cup squad and was substituted at half-time with slight calf trouble.

His replacement, Steven Naismith, was instrumental in giving Everton the lead in the 50th minute. After a corner by Baines was cleared to the edge of the area Naismith volleyed the ball towards goal and wheeled away in jubilation when it deflected off William Kvist and Stockdale before landing in the net. It was a freakish goal that left Fulham in need of a near-miracle.

Everton sought to compound the pain. Naismith sent Romelu Lukaku running free down the right and the Belgian forced a solid save from Stockdale. On the hour Baines burst into the opposing box but seemed torn between crossing and shooting and ultimately dabbed the ball just beyond the reach of Lukaku and wide of a post from 10 yards. Stockdale then had to show sharp reflexes to block a 12-yard blast from Lukaku.

Magath made a double substitution and one of the new arrivals, Ashkan Dejagah, delivered renewed hope in spectacular style, cutting in from the left before walloping a 20-yard shot into the net.

Fulham went for the win but again lacked the killer touch. Holtby clipped another dainty pass to Cauley Woodrow, whose close-range shot was beaten away by Howard, who moments later made another smart stop to thwart Pajtim Kasami.

Within moments Fulham had cause to lament those misses. Aiden McGeady dissected the home defence with a pass to Kevin Mirallas, who finished unerringly from the edge of the box. A neat close-range finish from Naismith three minutes from time left Fulham with a sinking feeling.

"It is very, very disappointing," said Magath, who did not even looke convinced himself as he suggested Fulham could still escape relegation. "It was a defeat but we played our best game. If we play like that again, let's see what happens."

Man of the match Lewis Holtby (Fulham)