Whether this precious victory is sufficient for Manuel Pellegrini to hold on to his job remains to be seen but West Ham held their nerve to record a second win since September. Sébastien Haller scored his first goal since October to hoist West Ham up to 15th, six points below fifth-placed Sheffield United. West Ham, who have been sounding out potential candidates to replace Pellegrini, could still dispense with the Chilean before Christmas.

That would seem cruel given the way West Ham performed here, dominating until a wobble late on when they survived a Southampton onslaught to seal victory. Asked whether he expected to be in charge against Crystal Palace on Boxing Day, Pellegrini was diplomatic. “You never know, you never know,” he said. “I don’t think that your job depends on whether you win or you lose, I think it depends on the trust of the owners of the club and I’m absolutely sure that they trust in what we do, but I cannot answer that question.”

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After Haller scored he ran to celebrate with the substitute Issa Diop while his manager punched the air. “I don’t know exactly if the word is relief,” Pellegrini said. “I’m always talking with the players that we must play the same way we work every day of the week, we must try to be a solid team and score goals and I think we are able to do it better. We lost three important games at home, games which maybe are the difference between a very good season and a bad season that we are doing.”

Festive cheer has been in short supply at West Ham but this victory may bring Pellegrini a slice of respite. Pellegrini had pointed to his team’s performance in victory at Chelsea a fortnight ago as evidence that his players retain faith in his methods and it was difficult to argue against that defence after an encouraging display here. Michael Antonio, who spent a season on loan at Southampton a decade ago, was outstanding up front alongside Haller, who returned to the starting lineup in place of Felipe Anderson, who was absent through illness. Haller would have doubled his and West Ham’s advantage in the second half but for a fine stop by Alex McCarthy and Antonio, who bullied Southampton’s backline throughout, had a goal ruled out by the video assistant referee, Jon Moss, after replays showed the forward handled in the buildup.

Jan Bednarek and Jack Stephens never got to grips with Antonio. He displayed the kind of hunger that was lacking in Ralph Hasenhüttl’s side and, when Haller struck, it was no surprise that his partner proved the catalyst. Antonio burst beyond Bednarek before laying the ball off for Robert Snodgrass, whose sweeping first-time cross found Pablo Fornals at the back post. Fornals cushioned a header into the path of Haller, whose scruffy volley bounced in off McCarthy’s left post. “I think playing with Antonio works better for him, Antonio had a lot of pace and created space and movement,” Pellegrini said.

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Southampton head to Aston Villa, who are above them on goal difference, next weekend before visiting Chelsea five days later. The only notable downside for West Ham was the injury to David Martin late on. Pellegrini said his reserve goalkeeper sustained an almost identical hip-muscle injury to the one that has sidelined the No 1, Lukasz Fabianski, since the end of September. “Exactly the same [but] I hope not so big,” said Pellegrini, who confirmed doctors would assess the severity of the injury on Sunday.

Southampton were painfully slow out of the blocks and ultimately punished for a slovenly performance. Shane Long was guilty of squandering the best opening, failing to tuck home the rebound from an acute angle after Danny Ings rattled the crossbar. Moments later Ings curled a beautiful shot into the corner but the referee, Martin Atkinson, had blown his whistle for a generous foul on the substitute Andriy Yarmolenko.

Jack Stephens, who had roamed forward from defence, equally passed up a golden opportunity after the captain Pierre-Emile Højbjerg cut the ball back inside the box. West Ham’s defending became more fraught as the game drew to a frenetic climaxbut they clung on to leave Southampton bruised.

“It was a tough game. We were fighting, investing a lot but in the end we were not able to take something,” Hasenhüttl said. “We know that the relegation goes until May and, like last season, it’s about battling for staying in. Everybody knows that. We know what it’s about to stay in the league and we know that it’s a tough fight until the end and we will take this fight.”