Just when it was least expected but sorely needed, Wayne Rooney popped up with his first ever hat-trick for Everton, proving that timing, like class, is probably permanent. The 32-year-old even managed to score the third of his goals from his own half with one of the cleanest first-time strikes you will ever see, which must have convinced the watching Sam Allardyce there is life in the old dog yet.

Somewhat belatedly it also appears David Unsworth might be getting the hang of this caretaking business. When Allardyce takes over his seventh Premier League club he will do so with Everton five points clear of the relegation positions, Unsworth’s parting gift having been the second win of his eight matches in charge and perhaps even more remarkably the first clean sheet. Everton may not be quite as bad as a week of relentlessly bad publicity has suggested, though of course it is possible that West Ham are in even bigger trouble than anyone imagined.

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Rooney helped himself to a couple of first-half goals as a much-improved Everton took hold of the game, all the more impressive since the captain’s present interpretation of the No10 role involves spending much of the game in his own half. He doesn’t normally expect to score from there, but when the ball found him in the centre circle after Joe Hart had left his area to clear from Dominic Calvert-Lewin, his immediate and correct instinct was to send it back strong and true, high enough to elude two West Ham defenders but firm and low enough to stay on target.

By the time Ashley Williams had headed in a late corner Allardyce must have been wondering which particular fires he is supposed to be fighting. His first two league games are against Huddersfield at home and Liverpool away, which should give a clearer picture. West Ham at the moment, much to the chagrin of David Moyes returning to Goodison almost unnoticed, look as if anyone could beat them.

Facebook Twitter Pinterest Wayne Rooney scores from inside his own half. Photograph: BT Sport

Moyes is on the sort of six-month contract at West Ham that Allardyce successfully resisted when Everton first approached him. He wanted a permanent situation, Everton were initially reluctant, until a few more woeful results persuaded them they were in no position to argue.

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As a result Allardyce has won an 18-month deal, a sort of halfway house, which is not bad going for a 63-year-old who has been at more Premier League clubs than Harry Redknapp (five). It means Allardyce will have thrashed out more lucrative terms too, which must have been some small consolation as he sat in the stand surveying the minimal difference in quality between one of his former sides and his new one.

Jordan Pickford made the first save of the game after 15 minutes, though in truth it was probably more of a cross from Pablo Zabaleta. The full-back should never have been allowed so much space on the right, but after Cuco Martina needlessly conceded a throw when the ball was running dead anyway, Everton missed a succession of chances to clear and would have been embarrassed had Zabaleta’s cross found someone in the middle. It didn’t, there was no one in the middle, which is probably the reason why West Ham came to Goodison two points below the home side in the league.

The visitors’ problems deepened midway through the first half when Pedro Obiang whisked the ball off Gylfi Sigurdsson’s toe but only succeeded in sending Calvert-Lewin bearing down on Hart. The goalkeeper dived at the forward’s feet a fraction after Calvert-Lewin had poked the ball around him, inevitably conceding a penalty. Hart was equal to Rooney’s initial spot-kick, diving to his right to beat the shot away, but the ball ran loose and it was a simple matter for Rooney to follow up and head in the rebound into the open net.

If there was an element of luck about the way the home side took the lead there was absolutely nothing wrong with the way they fashioned a second, a well-constructed team goal that might come to represent a much-needed high point of Unsworth’s caretakership. Beginning on halfway near the right touchline, Aaron Lennon did well to both keep the ball in play and send Jonjoe Kenny up the inside channel with a neat flick.

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Kenny found Tom Davies making a diagonal run into space to continue the attack, and though West Ham had men in position to cut out his low cross, the ball found its way through to Rooney for a firm finish and a second goal of the night. Everton survived a scare on the stroke of the interval when Manuel Lanzini’s corner ran across the face of goal, but once again there was no one on hand from West Ham to take advantage.

The visitors looked a little livelier in the second half, Aaron Cresswell skimming a shot against Pickford’s bar and the disappointing Marko Arnautovic shooting tamely at the goalkeeper from a better position than he probably realised. They were rewarded with a penalty on the hour, after Williams’s sprawling challenge was adjudged to have brought down Diafra Sakho, but though the decision was debatable Pickford rendered discussion irrelevant with a fine diving save from Lanzini’s shot. Goodison erupted with relief, little aware that in terms of vintage Rooney, the best was still to come.