It was a lone voice but the noise and sentiment will not have escaped Everton’s power-brokers, Farhad Moshiri and Bill Kenwright, sitting in the front row of the Goodison Park directors’ box. “Taxi for Koeman” came the anguished cry. Other Evertonians laughed. Most headed for the exits. Arsenal then scored a fourth, then a fifth. Turning the screw.

“Write what you want to write,” bristled Ronald Koeman when, having decried the amount of negativity around an Everton team now languishing in the bottom three, he was asked what positives he could see in a comprehensive home defeat.

The highly paid manager had needed a show of conviction, of progress and most of all a result with the pressure intensifying and him no closer to a solution to the conundrum posed by this summer’s investment of almost £140m. His team got a sound beating instead as Arsenal marked Arsène Wenger’s 68th birthday with a first away win in the Premier League this season and a scoreline to help exorcise the trauma of their last visit to Merseyside, the 4-0 reverse at Anfield in August.

Wenger deployed Alexis Sánchez, Mesut Özil and Alexandre Lacazette in attack for the first time this season and was rewarded with a vibrant, expansive and dominant display. But for Jordan Pickford, Everton’s £30m goalkeeper and rare success in the summer transfer market, Arsenal would have registered seven or eight. “That was maybe our most complete performance of the season defensively and offensively,” the Arsenal manager said. “Alexis was on fire. Mesut was superb – agile, quick, intelligent and always at the service of the team with his passing. He is an exceptional player.”

As for those Manchester United rumours about Özil, Wenger said: “As players and manager we can only give one response and that is to be focused on the pitch.”

For Koeman the focus will inevitably shift towards his future as Everton manager. His team are adrift, without hope and direction. Their manager is scrambling around for answers and producing nothing, although Koeman’s case here was not helped by the 68th-minute dismissal of Idrissa Gueye. Arsenal were on the front foot long before Wayne Rooney rolled back the years – 15 years and three days to be exact – to open the scoring with another long-range strike against Arsenal.

The 32-year-old’s finish carried echoes of his memorable debut goal against Wenger’s then league champions, in 2002. Right foot, Park End, textbook technique and struck to the Arsenal goalkeeper’s right; Rooney reminded the watching Gareth Southgate of his enduring quality as he used Nikola Vlasic’s decoy run to move into space before sweeping a 20-yard finish beyond Petr Cech.

For Arsenal it was an unnecessary – but only fleeting – reminder of their defensive deficiencies. Granit Xhaka was easily dispossessed by Gueye in midfield to allow Rooney to take possession. Defenders then backed off the former England and Manchester United captain as he lined up the 201st league goal of his career.

It was a gift for Everton, who desperately needed a release after two wins in 12 matches and had been besieged until that point. Koeman’s latest selection and formation shuffle had him arrive at a 3-4-2-1 shape that soon morphed into a bizarre 3-3-3-1, with Gueye all alone in central midfield as Vlasic followed his natural inclination to attack. Özil and Sánchez prospered in the space either side of the stranded Senegal international.

The only surprise about Arsenal’s equaliser was the identity of the goalscorer. Pickford had foiled Aaron Ramsey, Sánchez and Lacazette superbly but arguably could have done better with Xhaka’s low 25-yard shot than to parry back into the danger area. In mitigation, the midfielder’s effort took a slight deflection off Gylfi Sigurdsson, who endured another poor game, and Everton defenders reacted slower than Nacho Monreal to the rebound. The Arsenal defender converted emphatically and the stage was set for another dispiriting afternoon for Koeman and his team.

Everton fell behind after a rare foray forward broke down in the Arsenal area. The visitors broke incisively. Sánchez, Goodison’s pantomime villain after demanding a second yellow card for Ashley Williams, carried the ball down the left and floated a delightful cross between Michael Keane and Phil Jagielka. Between the central defenders ghosted Özil, who gave Pickford no chance with a pristine header from six yards out.

Belief, already fragile, vanished from Everton in an instant and their task became more ominous when Gueye was sent off for clattering into Xhaka, his second bookable offence. The needless challenge was duly punished.

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Arsenal’s third stemmed from a careless pass by Vlasic to Hectór Bellerín. Xhaka released Özil down the right and, with no royal blue shirt near him, the Germany international strolled into the area and squared for Lacazette to beat Pickford with a convincing finish. It was the cue for Arsenal to start show-boating.

Ramsey converted a fine through ball from the substitute Jack Wilshere to extend the visitors’ lead and, though Oumar Niasse grabbed a consolation having dispossessed Cech, it was left to Sánchez to have the final say with a precise finish into the far corner in stoppage time. The final word on the pitch that is.Moshiri and Kenwright may determine otherwise if their patience in Koeman has evaporated.