It was an evening when Arsène Wenger was left with a particular regret. Olivier Giroud pulled up in the 76th minute and when the Arsenal striker shot his hand back to the hamstring on his right leg, it did not look good. Wenger suggested the forward would be out for three weeks – in other words, the whole of the congested festive programme.

It was an occasion when Arsenal got the job done. It was not pretty and the headline statistic showed there had been only one shot on target throughout the game. It was a scruffy effort from Danny Welbeck – entirely in keeping with the overall tone – but it made the difference.

David Moyes said that “you certainly wouldn’t call it an Arsenal goal” but nobody in red was complaining, least of all Welbeck. The West Ham United manager bemoaned the space that Mathieu Debuchy was allowed to find in the inside-right channel and when he headed Francis Coquelin’s chipped ball across into the danger zone, Welbeck leapt between Winston Reid and James Collins.

Moyes could not understand why neither of his defenders was able to clear, and the ball broke kindly for Welbeck. From close range and without making a clean contact he bundled it past Joe Hart.

Arsenal have bigger fish to fry and the evidence was provided by Wenger’s decision to give the night off to every member of his starting line-up from Saturday’s Premier League win over Newcastle United. The visit of Liverpool on Friday evening looms large.

But after the first whistle had gone on this Carabao Cup quarter-final Wenger wanted to progress just as much as his second-stringers needed to impress. He got his wish. It was hardly a vintage performance against a West Ham team who offered frustratingly little as an attacking force. But it was enough.

Wenger has never won the League Cup in any of its guises but if he is to reach the final he will have to navigate a punishing schedule. When the two legs of the semi-final are factored in Arsenal face nine matches over the next 35 days. It is a period, as Wenger pointed out, when he cannot afford to lose players.

Danny Welbeck celebrates after opening the scoring for Arsenal moments before half-time. Photograph: Dylan Martinez/Reuters

Coquelin was also forced off before the end but the manager reported he had suffered only cramp to a hamstring and he should be available for Liverpool. Giroud was the concern. “A hamstring is normally 21 days,” Wenger said. “It looks like he is out for Christmas. We will normally do a scan 48 hours after [the injury] and then we will have an idea of the grade.

“When you hear him the pain is quite big, but the grade is not always linked with the pain.”

West Ham had arrived with confidence, after taking seven points from an available nine in the league, but there was no single moment when they stretched their hosts. As such, it did not seem as if they departed with any regrets.

Asked to chase the game in the second half, the sum total of their efforts was a 64th-minute free-kick from Aaron Cresswell that curled well wide. The 7,000 travelling supporters celebrated having had a shot. West Ham did not have too much of the ball and when they did, they took poor care of it. That disappointed Moyes but not as much as the concession. “It was a horrible goal to give away,” he added.

Arsenal pressed on to the front foot but there was zero excitement until the 39th minute, when Sead Kolasinac made inroads up the left. He crossed and when Collins and Angelo Ogbonna followed Giroud to the near post, Theo Walcott was left unmarked in front of the penalty spot.

The home fans howled when he directed his diving header wide of a post. It was a long way from being Walcott’s night.

It is plain Moyes does not have the same resources as Wenger and he retained five of the starting XI that had won at Stoke City last Saturday. It was also easy to see this Saturday’s home game against Newcastle is his priority.

Moyes made his move with the double introduction of Andy Carroll and Diafra Sakho. The plan B was simple: hit Carroll with the high ball. It did not work. Hart was booked after he mistimed a challenge on Welbeck in the 71st minute just outside the area, and West Ham never threatened the equaliser.