For virtually the entire campaign Huddersfield, overwhelming pre-season favourites for relegation, have fought valiantly to keep their heads above water. But the risk of them sinking has always been present and it increased with this result.

David Wagner’s team are still clear of the bottom three but goals by Cenk Tosun and Idrissa Gueye, combined with results elsewhere, mean that Wagner believes his team will reach safety only if they scrounge points from a daunting final week of the season in which Huddersfield take on a trio of giants.

“I think 35 points will not be enough to stay up,” said Wagner, whose side face trips to Manchester City and Chelsea before hosting Arsenal in a match that Huddersfield hope will not be their last in the Premier League, even if it will be Arsène Wenger’s. “Last year we worked our socks off to play these opponents and now we play them in the last week of the season and have to get points. For two and a half years we have worked on the impossible and now we face a week that is very difficult but success is possible. Other teams have shown it is possible to steal points off the top six and this is our task now.”

Wagner had hoped to make that task unnecessary by beating Everton. “It absolutely feels like a missed opportunity,” he admitted. “I said to the players before: ‘Is it a final?’ Yes, because we are able to get over the line [by winning] but on the other hand, maybe it isn’t, because we will have three more opportunities afterwards, admittedly against top teams.”

They began their maybe-a-final warily and never looked like producing enough sharpness to beat Everton. That has been a recurring problem during a league campaign in which they have failed to score in 19 of 35 matches.

As for the visitors, many of their supporters still want to see the back of Sam Allardyce – sporadic chants from the 2,300 travelling fans made that much clear – but the manager maintains he is not going anywhere and hopes this victory, on top of positive recent results, will help fans warm to him.

“I can’t produce more than I’m doing,” he said. “What can I say when we’ve got 14 points from the last seven games when the club was in a very difficult position when I arrived? I can only say it’s one of those things. If we keep winning, we might change it to love.”

There was not much to love or fear about the visitors early on. Everton had a lot of the ball but advanced with the joy and pizzazz of a chain gang. One could understand why Evertonians complain that watching their team can feel like serving a sentence.

Huddersfield were emboldened by Everton’s impotence. Steve Mounié landed the game’s first shot on target in the 17th minute but Jordan Pickford saved easily.

Three minutes later Terence Kongolo tried his luck from 20 yards but his shot looked like affecting the scoreline only in that it nearly struck the giant scoreboard at the top of the stand. Rajiv van La Parra got a little closer on the half-hour when he drove the ball a couple of yards over the bar after a pull-back by Alex Pritchard.

Everton were shrinking all over the pitch and perhaps their dwindling was what lulled Van La Parra into a costly lapse in the 39th minute. The winger sent an attempted crossfield pass straight to Theo Walcott, who released Tosun. Play seemed to unfold in slow motion as the Turk lumbered forward and directed a low cross-shot from the right-hand corner of the box into the bottom corner of the net. It was Tosun’s fifth goal since joining Everton in January and, for Huddersfield, a sickening strike against the run of play.

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Everton improved in the second half, pre-empting any Huddersfield fightback, and Jonas Lossl was forced to make good saves from Morgan Schneiderlin and Séamus Coleman. Gueye confirmed the visitor’s win in the 77th minute by sweeping a low shot into the net from 16 yards after a simple tee-up by Leighton Baines.