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David Moyes had stressed before Fulham's visit that the responsibility for scoring goals should not fall solely on the team's strikers. “I expect more goals from my centre-halves and midfielders as well,” he insisted. The Senegalese midfielder and Michael Keane responded perfectly, delivering a fully deserved victory over Marco Silva’s ineffective side.

The Merseyside club's second win in nine matches was fairly straightforward as the visitors showed the reason their leading scorer this season is goals gifted by opponents. Aside from a short spell in the second half, the visitors were subdued throughout by the home team's greater urgency and quality. Moyes’ team had three goals ruled out for infringements, but a close-range strike from Gueye in first-half stoppage time and the defender's second-half header ensured there would be no comeback for their ex-coach.

No one needed a goal more than Thierno Barry, the Everton forward who had gone 10 Premier League outings without testing the goalkeeper after his £27m summer arrival from Villarreal and spurned a gilt-edged chance to put his team 2-0 up at the Stadium of Light on Monday. The youngster headed the first opportunity of the game wide of Bernd Leno’s crossbar when found by Iliman Ndiaye’s excellent delivery.

The home side controlled the opening stages and the Fulham goalkeeper pushed over James Garner’s long-range set-piece, awarded after Sasa Lukic was booked for hauling down the Everton midfielder. The Serbian brought down the identical opponent later in the half but the official, Andrew Madley, correctly waved away home protests for a second yellow. The Fulham boss was taking no further chances, however, and withdrew the player at the interval.

The striker believed his luck had changed at last when sliding in at the far post to convert a low cross by Gueye. But the elation of a maiden strike was erased by an assistant referee’s flag. Ndiaye was offside when attacking the delivery, and failing to connect, and the VAR backed up the original call. Barry’s misfortune may have continued in the final third, but his overall display justified Moyes’ decision to keep the faith. His runs and work-rate occupied Fulham’s central defenders and contributed to Everton the upper hand all game.

Fulham grew into the game slowly with Sander Berge and the ex-Goodison player the Nigerian working well in the engine room, but the first half threat from the away team was minimal. Raúl Jiménez shot tamely at the England keeper when teed up inside the area by Iwobi and sent a free-kick from a dangerous position directly at the Everton wall. And that was it.

Everton, inspired by the midfielder and Ndiaye, had a another strike disallowed for an infringement when the Fulham goalkeeper saved a effort from Keane and the captain fired home the rebound. The skipper had moved beyond the last defender when heading on Jack Grealish’s cross in the build-up. But Everton’s third attempt past Leno counted. The left-back floated a perfect ball to the back post when left unmarked on the left by the youngster. Tarkowski connected with a thumping header against the bar and, though Iroegbunam mishit the rebound, his teammate Gueye finished from point-blank. The sense of release inside the ground was evident.

Everton had a further effort disallowed after the restart after Dewsbury-Hall scored from a further excellent Mykolenko cross. Ndiaye had cushioned the delivery into the striker, who was offside when challenging Joachim Anderson for the ball that fell to the home player. Everton would have to wait until the 81st minute for the security of a two-goal lead. The provider was the architect with a corner that Keane directed past Leno. He did so with the upper body, and Fulham’s appeals for a handball were dismissed by the video official.

Fulham carried more of a threat after the substitutions of Josh King, Rodrigo Muniz and Adama Traoré. The Everton keeper made a fine stop with his feet to deny Muniz finding the net with his initial involvement and denied the speedster with another important stop late on.