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David Moyes had made clear before Fulham's visit that the onus for scoring goals should not fall solely on the team's forwards. “I expect more goals from my defenders and central players as well,” he insisted. The Senegalese midfielder and the English defender responded perfectly, earning a merited victory over Marco Silva’s toothless side.

The Merseyside club's second victory in nine matches was relatively comfortable as the visitors highlighted why their top marksman this season is goals gifted by opponents. Apart from a brief flurry in the latter period, the visitors were subdued all match by the home team's superior intensity and quality. The Blues had three efforts ruled out for offside, but a close-range strike from Gueye in first-half stoppage time and Keane’s second-half header made sure there would be no comeback for the former Everton manager.

No one was more in need of scoring more than Thierno Barry, the Everton forward who had failed to register a shot on target in 10 league games without testing the goalkeeper after his £27m summer arrival from the Spanish side and spurned a clear opportunity to put his team two goals ahead at Sunderland on Monday. The 23-year-old directed the earliest chance of the game over the Fulham keeper's crossbar when picked out by Iliman Ndiaye’s fine cross.

Everton dominated the early exchanges and the visiting shot-stopper pushed over James Garner’s long-range set-piece, awarded after Sasa Lukic was booked for hauling down the Everton midfielder. Lukic tripped the identical opponent later in the half but the official, Andrew Madley, correctly waved away home protests for a sending off. Silva was not risking anything, however, and substituted the midfielder at the interval.

Barry believed his fortune had finally turned when sliding in at the far post to turn in a drilled pass by his teammate. But the elation of a maiden strike was erased by an assistant referee’s flag. Ndiaye was offside when going for the delivery, and missing, and the video assistant referee backed up the on-field decision. Barry’s misfortune may have continued in the final third, but his overall display justified Moyes’ decision to stick with him. His movement and effort occupied Fulham’s central defenders and contributed to Everton the edge all game.

Fulham grew into the game gradually with Sander Berge and the former Everton midfielder the Nigerian working well in the engine room, but the first half threat from the away team was minimal. The Mexican striker shot tamely at the England keeper when teed up in the box by his teammate and put a set-piece from a promising location directly at the Everton wall. And that was it.

Everton, inspired by Dewsbury-Hall and the forward, had a second goal disallowed for offside when Leno parried a Keane header and James Tarkowski fired home the loose ball. The home captain had moved beyond the last defender when nodding down the winger's delivery in the build-up. But the team's third attempt past the keeper counted. Vitalii Mykolenko floated a perfect ball to the back post when found in space on the left by the youngster. Tarkowski connected with a thumping header off the crossbar and, though the midfielder fluffed his lines, his midfield partner the scorer converted from point-blank. The sense of release inside the ground was evident.

Everton had a further effort ruled out after the restart after Dewsbury-Hall scored from a further excellent delivery from the left. The attacker had cushioned the delivery into Barry, who was offside when challenging Joachim Anderson for the touch that reached the Everton midfielder. The team would have to wait until the 81st minute for the comfort of a second goal. Dewsbury-Hall was the creator with a set-piece that the defender directed past Leno. He scored with the upper body, and Fulham’s appeals for handball were dismissed by VAR.

Fulham carried more of a threat after the substitutions of Josh King, the Brazilian and Adama Traoré. The Everton keeper saved well with his feet to deny the substitute scoring with his initial involvement and stopped Traoré with a crucial save late on.