By the time the evening temperature plunged towards zero and Tyneside pavements started turning icily treacherous, Newcastle United had already lost their footing.

Undone by Wayne Rooney’s first-half goal, Rafael Benítez’s suddenly unsteady side were heading for a seventh defeat in eight games. This latest reverse featured some improved performances but Newcastle have now taken only one point from a possible 24 and Benítez clearly did not relish the smile which adorned Sam Allardyce’s face at the final whistle.

By way of exacerbating the Spaniard’s problems, the recalled Jonjo Shelvey once again failed to merit his manager’s trust. Shelvey’s enduring lack of judgment seemed highlighted when he was sent off for a bad tackle on Idrissa Gueye in the 94th minute which resulted in his second yellow card of the night.

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Benítez was suitably disappointed. “We had the game under control but we made a mistake and then the way Everton were wasting time at the end made things really difficult,” he said. “We worked so hard and it’s upsetting but we all, including me, have to take responsibility.”

In contrast Allardyce presided over Everton’s first away win in the league since January. “A defensive performance which kept Newcastle pretty quiet pleases me very much,” said Ronald Koeman’s successor, whose presence seems to have instantly improved fortunes at Goodison Park. “We move onwards and upwards.”

Ironically, Newcastle’s day had begun on a rare high. The pre-match news that Mike Ashley and Amanda Staveley are close to agreeing a price for the sale of the club ahead of a potential takeover being completed at some point in January or February certainly created an air of buoyancy in the stands.

Flattering to deceive, Benítez’s players began in similarly upbeat mode. As Everton struggled to assume possession, let alone string a few passes together, and Newcastle parked their tanks in Allardyce’s penalty area, home fears of a looming relegation battle temporarily appeared overblown.

Given his long-standing personal rivalry with Benítez – not to mention the small matter of an unhappy eight months in charge here during 2007 and 2008 – this was one fixture Everton’s manager was particularly keen to win, though.

If Allardyce’s increasingly intense technical area gum chewing suggested a degree of early concern, he was not about to panic. For all Newcastle’s huffing and puffing, the excellent Jordan Pickford was rarely troubled and, almost imperceptibly, Everton began both seeing more of the ball and using it fairly sensibly.

They weathered an odd scare – most notably when a counterattacking, thoroughly rejuvenated Matt Ritchie hit a post with a blistering left-foot shot – but took the lead after Rooney capitalised on Karl Darlow’s goalkeeping error.

Minutes earlier an unmarked Dominic Calvert-Lewin had been unable to quite connect with Gylfi Sigurdsson’s inviting cross but the striker redeemed himself by crossing for Aaron Lennon to test Darlow’s reflexes with a header.

The goalkeeper touched the ball and should have held it but instead saw it slip tantalisingly from his grasp, permitting Rooney to slide in, stretch out his right foot and stab home from point-blank range. It was Rooney’s 15th Premier League goal against his favourite opponents.

Newcastle should have equalised when Mikel Merino unleashed a fabulous left-foot shot which rebounded off the inside of a post but, generally, Everton’s off-ball positioning restricted Benítez’s side to half chances.

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The Tynesiders were playing considerably better than in recent weeks but although Mason Holgate was forced to scoop Merino’s goal-bound shot to safety they could not sustain a ferocious early second half tempo.

After reverting to a single striker system at kick off, Benítez replaced Mo Diame with Joselu but Everton subsequently came closer to scoring when Darlow did well to turn Ashley Williams’s header round a post following Sigurdsson’s fine delivery.

With Everton’s impressive left-back Cuco Martina largely nullifying Christian Atsu’s pacy right-sided threat and the substitute Nikola Vlasic proving highly effective at knitting visiting counterattacks together, Newcastle started to run out of ideas as their guests increasingly killed time. Then Shelvey got his marching orders. “I’m pretty pleased,” said Allardyce. “We’re rolling forward.”