It will take more than victory over an abject Bournemouth team to blow the storm clouds away but for Arsène Wenger and Arsenal, this was a start. Inspired by Danny Welbeck, they took what was a must-win game by the scruff of the neck and never looked like suffering further early-season damage.

There had been plenty of time for Arsenal’s edgy fan base to pick over the humiliation from Liverpool two weeks ago and the tension had been simmering. Yet Bournemouth were the most obliging of opponents and they remain pointless for the season – which was a neat way of reflecting upon their display.

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Wenger had named Alexis Sánchez among the substitutes, following his return on Thursday from two World Cup qualifiers with Chile – which was not a surprising move – and it meant that Welbeck was able to retain his starting place. He more than repaid Wenger’s faith, putting Arsenal in charge with a scruffy early finish, adding an assist for Alexandre Lacazette and another, altogether cleaner strike in the second half.

Sánchez had been invited onto the field before the game to receive his Supporters’ Player of the Season trophy for his efforts last time out. He smiled for the photographers and then legged it up the tunnel. He would be introduced from the bench in the 75th minute and the reaction from the home support was best described as mixed. There were boos from some fans – a reaction to Sánchez’s attempt to force a move to Manchester City before the transfer deadline.

Arsenal played the final 10 minutes with 10 men, after another of their substitutes, Francis Coquelin, limped off with a hamstring injury; Wenger had made all of his changes. It was the prompt for Bournemouth to finally work Petr Cech through Charlie Daniels and Josh King but it was too little, too late.

Wenger acknowledged the boos for Sánchez and he made the point that “the best way to get people on your side is to perform”. He was happier to talk up Welbeck, who was a dynamic presence from the outset. “Welbeck gets stronger and stronger,” Wenger said. “People are not convinced that he is a great finisher but these kind of goals will help him.”

Eddie Howe had spoken about the importance of a good start for Bournemouth but they were behind after six minutes, following the failure of their three-man backline to track Welbeck. It was a sharp piece of movement from the Arsenal forward inside the area and he will argue that he made his own luck when he rose to meet Sead Kolasinac’s inviting centre. The ball squirmed in off his shoulder, rather than his forehead.

Welbeck was involved in the second goal, although Bournemouth will lament the part that Nathan Aké played in it. The defender moved over to deal with Mesut Özil’s ball forward but he swiped at it and missed. It ran for Welbeck, who turned it back to Lacazette and the £53m summer signing – in the space vacated by Aké – guided a glorious curler up and around Asmir Begovic.

Bournemouth were horribly loose with and without the ball and their travails were summed up in the 37th minute when Begovic took a heavy touch following Tyrone Mings’s backpass and was closed down by Welbeck. The ball broke towards Lacazette but Begovic reacted quickly to block at the striker’s feet. The goalkeeper directed a tirade at Mings, in which he essentially told the defender that he did not want the ball like that.

Howe squirmed during the first half, when his team barely crossed halfway. They left spaces at the back, with Lacazette preying on those in between the central defenders, while in possession they were – to use Howe’s words – wasteful, negative and slow.

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The half-time whistle might have felt like a relief and Arsenal could conceivably have been further in front. Begovic saved smartly from Granit Xhaka and Özil while Xhaka lifted another shot over the bar and Kolasinac saw a volley blocked by Adam Smith.

Howe still knew that if his side could pull a goal back it could see Arsenal wobble and they almost got it on 48 minutes when Jermain Defoe headed the substitute Jordon Ibe’s cross against the post. Bournemouth, reconfigured into 4-3-3, with Aké having stepped up into midfield, had a flicker of hope. But it was snuffed out almost immediately when Welbeck got his second.

Once again, the goal was marked by Bournemouth sloppiness. Dan Gosling dwelt in possession and he was hassled by Lacazette and robbed by Aaron Ramsey, who drove at the visitors’ penalty area. Ramsey ushered in Welbeck, who shot low and first time across Begovic and into the net.

Welbeck almost had the hat-trick only for his chip to drift inches wide while Coquelin and another substitute, Olivier Giroud, went close. Arsenal have their respite.