Manager Arsene Wenger says Arsenal cannot afford further injuries
But England forward Danny Welbeck broke West Ham's resistance just before the break, turning in Mathieu Debuchy's header across goal as the hosts stayed on course for a first League Cup win since 1993.
The Gunners' night was soured by the loss of Giroud to an injury that could rule him out for three weeks, while Francis Coquelin also limped out of the match late on.
With Arsenal embarking on a run of 10 games in 36 days, Wenger admitted the loss of Giroud had come as a blow.
"It is very demanding, especially because we already have a tough schedule behind us," he said. "We have to sit down and analyse what we can do.
"What is very worrying is when you start to lose players, like Giroud tonight. After, you cannot afford to lose another one. I have to look how I can manage the schedule.
"When you look at our fixtures in January and February, you cannot imagine we will always play with the same players in every game.
"Will I change from Premier League game to Premier League or only in the Carabao Cup? I don't know yet - I have to look at the fixtures to see what I do."
West Ham boss David Moyes also named a much-changed side but they failed to click in attack, with the belated introductions of Andy Carroll, Diafra Sakho and Marko Arnautovic unable to undo the home defence.
Neither Joe Hart, restored to the West Ham side for the rested Adrian, nor Arsenal's second-choice goalkeeper David Ospina were forced into much action in what was a disjointed contest.
Moyes' side had been content to sit back and try to catch Arsenal on the counter during a low-quality first half, but even with the attacking changes they could not muster a single shot on target.
Welbeck's goal aside, Arsenal carried little threat to Hart's goal with Theo Walcott spurning their only other clear-cut chance when he headed wide from 10 yards.
Walcott's poor miss, when he had time to bring the ball down and really test Hart, summed up his contribution on a night when Wenger had handed him a chance to impress.
The 28-year-old could have used his inclusion force his way into his manager's Premier League plans but in the end his performance came up short.
"There has been talk of him leaving but is he trying to prove himself and stay at the Emirates? Tonight he hasn't really impressed and I wouldn't say he's good enough to get in Arsenal's starting XI at the moment."
It was only Walcott's third game in almost two months after illness and injury, so a little rustiness can be forgiven, but he will need to do more than this if he wants to force his way back into the first-team reckoning.
Rotating line-ups is certainly nothing new in this competition, but even so Wenger took something of a risk by making wholesale changes with one eye firmly on Friday night's important Premier League match at Liverpool.
Amid the upheaval, the Arsenal manager will have been satisfied to note the composed performance of 18-year-old Joe Willock in midfield and by how well defender Calum Chambers - on just his fourth appearance of the season - nullified West Ham striker Javier Hernandez and late replacements Sakho and Carroll.
Nineteen-year-old Ben Sheaf was also afforded a run out as he made his domestic debut for the Gunners.
The changes no doubt contributed to a poor game, but the Gunners still got the job done as they booked a place in the last four of the League Cup for the first time since 2011.
For their part, West Ham made six changes from their impressive 3-0 win at Stoke, but a Hammers midfield featuring 18-year-olds Declan Rice and Domingos Quina struggled to retain the ball and contributed to the visitors' lowly 33% possession statistic.
Moyes' overriding aim at West Ham is to keep the club in the Premier League, and in that regard things are moving in the right direction with seven points from a possible nine in recent weeks.
But for a manager who has never won a major trophy in his 19-year managerial career to date, this felt like a missed opportunity.
Had his side taken advantage of much-changed and ill-at-ease opposition, they were just three games away from possible silverware, but as it was the Hammers sat deep, leaving lone striker Hernandez to feed off scraps.
Moyes made a double attacking change with 25 minutes left, bringing on Carroll and the in-form Sakho.
The pair put a little more pressure on Arsenal's backline but shots on target remained utterly non-existent.
Arsenal manager Arsene Wenger: "I felt we played with quality and spirit and pace in the first half. Overall for 90 minutes we looked solid defensively and weren't tested at the back too much. Tested, but not in danger.
"Second half we lost a bit of pace and were less creative and then couldn't the game off. So it was down to not making a mistake.
"Overall it's a positive night for us, other than we lost Olivier Giroud. Francis Coquelin looks less problematic, but Giroud looks to be out for Friday night."
Arsenal host Liverpool in the Premier League on Friday (19:45 GMT), while West Ham welcome Newcastle to London Stadium on Saturday (15:00 GMT).
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