David Moyes said his players “have to keep stepping up” after West Ham ended the perfect week with a vital 4-0 victory at Bournemouth to boost their survival chances. Moyes’s side fought back from two down to draw with Arsenal, the Premier League leaders, last Sunday and followed it up with a 4-1 win against Gent on Thursday to book their spot in the Europa Conference League semi-finals. Relegation has remained a distinct possibility, though, but they eased those concerns with a clinical display where first-half goals by Michail Antonio, Lucas Paquetá and Declan Rice put them in control. Pablo Fornals, on as a substitute, added a fourth midway through the second half to help West Ham to move up to 13th and six points clear of the bottom three. Despite their Thursday exploits, Moyes made only two changes at the Vitality Stadium and one was to recall his first-choice goalkeeper, Lukasz Fabianski. It meant Antonio got another opportunity to lead the line and he rewarded his manager’s faith with a fifth-minute opener. For Bournemouth it was all too easy, with Aaron Cresswell’s near‑post corner powered home by the unmarked Antonio for his 12th goal of the season (though only his fourth in the league). The visitors doubled their advantage eight minutes later and, again, it must have frustrated Gary O’Neil, the Bournemouth manager. Vladimir Coufal won possession from Marcus Tavernier and controlled Jarrod Bowen’s pass before his looping cross was headed in at the back post by Paquetá, who easily outjumped Jack Stephens. It continued Paquetá’s encouraging form since the World Cup break and backed up his goalscoring display in the Hammers’ triumph last Thursday. West Ham’s travelling support were enjoying their trip to the south coast and soon singing about rivals Tottenham, who were 5-0 down at Newcastle inside 21 minutes. Bournemouth were not prepared to throw in the towel, however, and Chris Mepham tested Fabianski with a header moments later. Jefferson Lerma was next to try his luck and Fabianski was again equal to it, but his best stop was to come. Dominic Solanke muscled his way past Nayef Aguerd and raced into the area but Fabianski saved well with his feet to preserve West Ham’s two-goal lead after half an hour. Having withstood Bournemouth’s fightback, Moyes’s team then provided a sucker punch two minutes before half-time through their midfield talisman. Another Cresswell corner caused problems and, while it was cleared to the back post, Rice was first to the loose ball and on hand to drill home via a deflection to make it back-to-back goals this week. The West Ham captain was serenaded by the away fans soon afterwards, with chants urging him to stay amid talk his future could lie away from the London Stadium. O’Neil responded with the Wales forward Kieffer Moore introduced in place of Joe Rothwell at half-time, but it failed to have the desired effect. A triple substitution followed on the hour for Bournemouth with Dango Ouattara, the stoppage-time hero at Spurs last weekend, brought on but, while he initially caused problems, West Ham were soon celebrating a fourth goal. Bowen did well down the right and, while his cross was slightly behind Fornals, the substitute managed to back-heel into the far corner to make it 4-0 after 72 minutes. Maxwel Cornet had a stoppage-time effort ruled out for offside and the West Ham fans finished the game still chanting for Rice to stay one more year, with this win at least going a long way to confirming their place in the Premier League next season. Asked about Rice after the game, Moyes tempered his praise with constructive criticism. “I told him that I didn’t think his passing had been good enough in the opening part of the game and that is because his standards are so high,” he said. “But I am not just going to pat him on the back all the time, too much praise makes you soft. I have been saying it for a long time, he’s a top player. But look, I can’t say too much because I talk so glowingly about him and I want to make sure the other players realise they are as important, if not more important. They have to keep stepping up, we all have to keep stepping up to make sure we keep going.” Bournemouth, now 15th in the table, will aim to regroup quickly for the trip to Southampton, their south-coast rivals, on Thursday. “We obviously dominated the ball but if you watch West Ham they have real strengths,” O’Neil said. “We weren’t able to cope with them, basically.”