Giroud's superb strike early in the second half helped secure a first win in five Premier League outings for the FA Cup finalists, who are now a point ahead of Everton in the race for the last UEFA Champions League berth, but have played a game more.
Matt Jarvis had nodded the Hammers into the lead late in the first half, but that appeared to rouse Arsene Wenger's men from a sluggish display and Podolski levelled before the interval, with the Germany international adding a third 12 minutes from time.
Giroud had a good chance to break the deadlock with 14 minutes gone, but there was a lack of conviction in his attempt to get on the end of Podolski's cross on the edge of the six-yard box and the opportunity went begging.
West Ham had started strongly and came close to finding the net twice in the space of 60 seconds, with Thomas Vermaelen forced to head Jarvis' deep cross away under pressure from Andy Carroll, before Mohamed Diame slashed over the bar when well-placed from the resulting corner.
Giroud then missed a glorious opportunity when he was played in behind the Hammers defence by Santi Cazorla, with his ponderous touch and attempted flick with the outside of his boot making it too easy for Adrian to gather.
West Ham made the Gunners pay five minutes before the interval. Antonio Nocerino was not tracked by Kim Kallstrom as he burst into the box and, although his initial effort was saved, the ball looped up for Jarvis to nod home bravely, with Laurent Koscielny's flailing boot only narrowly missing his head.
Arsenal responded and Kallstrom forced Adrian to save at his near post before Podolski levelled with a drilled low finish across the keeper after Cazorla seized onto Stewart Downing's poor clearance and threaded a pass through the defence.
Downing fired a low shot wide early in the second period before Diame got away from a weak Mikel Arteta challenge far too easily only to balloon his long-range effort way off target
Giroud then made amends for his earlier misses in stunning style, bringing down Vermaelen's deep ball to the back post with a deft touch from his left boot and then firing beyond Adrian with his right despite the close attentions of Winston Reid and Carroll.
Arsenal were keen to press home their advantage and Adrian was forced to fist away Podolski's powerful free-kick from the right corner of the box, before Cazorla cut inside after a swift Arsenal break with his shot curling away from the keeper but also inches beyond the far post.
Aaron Ramsey, who suffered a thigh injury in the win at West Ham on Boxing Day, was sent on in place of Tomas Rosicky for his first Emirates appearance since late December in the 72nd minute and he lashed a long-range shot over the bar with his first meaningful contribution soon after.
The Wales international then played a key role in Arsenal's third, with his fine cushioned header from Giroud's cross releasing Podolski to fire over Adrian from 10 yards out and make the points safe for the Gunners.