Romelu Lukaku got the goal he probably expected and Ross Barkley the goal he definitely needed as Everton signalled an important step in their development under Ronald Koeman. A 2-0 lead against West Ham United was the cue for Everton to implode at Goodison Park last season. Only Slaven Bilic’s defence collapsed this time out.

Everton found relief not only in a first win in six matches but in the contribution of Barkley, who was involved in the breakthrough, started and finished the move that sealed victory and was denied another only by a superb save from Adrián. It was a timely response to criticism of his end product and talk of Wayne Rooney returning to Goodison, both emanating from Koeman himself.

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“It was one of his best games of this season,” the Everton manager said. “He was working hard, he was tackling offensively, he was important in the game and Adrián made an incredible save in the first half and then he scored the second goal. He was really the player behind the defence and the front to create.”

Goodison also witnessed the contrast between Koeman and his Everton predecessor as West Ham paid a substantial price for defensive lapses and repeated failures in front of goal. With seven minutes remaining Koeman replaced a winger, the influential Yannick Bolasie, with the defender Phil Jagielka to close out an entertaining contest.

Last season Roberto Martínez replaced a winger, Aaron Lennon, with the striker Oumar Niasse in the 76th minute with Everton leading 2-0. West Ham responded with three goals and Martínez’s relationship with the Goodison crowd never recovered. There would be no repeat as Everton, marking the one-year anniversary of Howard Kendall’s death, rediscovered their cutting edge and the way to win.

Facebook Twitter Pinterest Romelu Lukaku celebrates scoring with Kevin Mirallas. Photograph: Carl Recine/Reuters

It proved a torturous afternoon for Bilic. West Ham started with the confidence befitting a team with three straight victories and improved attacking options but their threat and concentration deteriorated after the interval as Everton delivered a lesson in ruthlessness. Lukaku opened the scoring with his ninth goal in 11 appearances against the Hammers and, with the hosts sharp on the counterattack, the visitors’ pursuit of an equaliser left them exposed.

“Harsh? No,” was the verdict of the West Ham manager. “They were more lethal or more mature or more cruel. When they were offered something, they took it straightaway which we didn’t do when we created or they made mistakes. There is a bitter side because we should take something out of this game. I had a feeling before the game and at half-time there was something for us. We played some good football, created some chances and I felt they were really vulnerable at the back, especially at the beginning of the game, but then we gave a goal away. But not harsh, I can’t blame anyone other than us. It is not the referee or bad luck. It was all 50-50. We just didn’t take it.”

Pedro Obiang squandered a glorious chance created by Dimitri Payet early on and the latter as well as Michail Antonio and Manuel Lanzini had sight of Everton’s goal in the first half but were unable to convert. Joel Robles, deputising for the injured Maarten Stekelenburg, also deserves credit for his clean sheet having saved well from Antonio and the captain, Mark Noble.

Everton opened nervously but improved as the game wore on. Barkley thought he had opened the scoring when he collected Bryan Oviedo’s cut-back on the left, stepped inside Winston Reid and shot powerfully from 12 yards. The ball took a deflection en route yet Adrián somehow read the flight to tip over with a superb reflex save. Reid had controlled Lukaku impressively in the first half but their role reversal was instrumental in deciding the outcome in the second.

The Belgium international converted his inevitable goal against West Ham due to a series of defensive errors coupled with quick-thinking by Bolasie. Reid put his team under pressure with a weak clearance that enabled Barkley to set up Seamus Coleman for a low drive that Adrián parried to his left. Reid and Aaron Cresswell each left the loose ball to the other and Bolasie seized on the indecision to slide in and clip the ball back across goal. Lukaku, standing onside and in front of an open goal, nodded home.

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West Ham were re-energised by the introduction of André Ayew and almost level when Ashley Williams slipped under pressure from Antonio, who was denied by Robles. Antonio’s two team-mates, better placed inside, were less than impressed.

The visitors’ pursuit of an equaliser left them open to Everton’s quality on the break and Barkley started and finished a flowing move when he spun away from two challenges in the centre circle and sprayed the ball down the right channel for Lukaku. The striker cut to the by-line and floated a perfect cross to the back post where Barkley had arrived unmarked and beat Adrián with a precise half-volley into the bottom corner.