The Frenchman scored an excellent opening goal before playing his part in strikes for Aaron Ramsey and substitute Mathieu Flamini as Arsene Wenger's men march on.
West Ham battled valiantly despite being under the cosh for much of the first half, before Giroud justified his inclusion ahead of Monday’s FA Cup hero Danny Welbeck when he finished off a well-worked move shortly before the interval.
The Hammers gave Arsenal some awkward moments in the second half too, but Giroud combined with Aaron Ramsey for the Welshman to double the advantage nine minutes from time and then played in Santi Cazorla, who in turn laid on the third for Mathieu Flamini.
In the end, Arsenal eased to a 10th consecutive victory over West Ham in all competitions while it also built further confidence as the sde seeks to overturn a 3-1 first-leg deficit in the Champions League against Monaco on Tuesday.
West Ham boss Sam Allardyce was forced into naming a makeshift center back pairing of Cheikhou Kouyate and James Collins due to injuries.
Theo Walcott, meanwhile, was one five Arsenal changes from that cup win over Manchester United, and he should have opened the scoring following Giroud's neat back heel early on.
Walcott hesitated, allowing Collins to snuff out the danger, before former Arsenal man Alex Song denied Alexis Sanchez a shot on goal with a well-timed challenge.
The home team's early onslaught continued, as Alexis' header was saved by Adrian before Kevin Nolan had West Ham's first effort on target - volleying at David Ospina following positive play down the left from Matt Jarvis.
That sparked a good spell for Allardyce's side, although Ramsey - another to come into Wenger’s XI - almost beat Adrian with an improvised effort off his knee as Arsenal started to regain the upper hand.
After several wasted Arsenal chances, Giroud finally rewarded his side’s persistence in first-half stoppage time when he capped off wonderful interplay with a fierce left-footed strike.
Despite that blow, West Ham started the second half strongly, as Diafra Sakho almost connected with a teasing delivery from the left before Jarvis fired a loose ball over the crossbar.
At the other end, the resistance of Collins and Kouyate gradually began to fade, and late on Giroud provided the inspiration again to make the result more comfortable.
First, he was on hand to play in Ramsey, who produced a smart finish inside the area for the second goal.
The Frenchman then clipped the ball in for Cazorla this time to find substitute Flamini for a close-range finish six minutes from time, ensuring Arsenal's 10th consecutive home win in domestic competition.