After a slow start following their nerve-shredding penalty shoot-out FA Cup win over Wigan Athletic at Wembley on Saturday, Arsenal's first big chance arrived - on the half-hour mark - when Santi Cazorla latched on to a stray pass in midfield. He then fed Giroud, who beat the offside trap, but his scuffed finish was saved at full stretch by Adrian.
It proved costly as West Ham went ahead when Jarvis headed in from close range after Antonio Nocerino's shot was saved at the near post.
Arsenal, though, were level just before half-time as Podolski spun in the West Ham penalty area to drill a low shot into the bottom corner.
Giroud finally showed the perfect touch to put Arsenal ahead 10 minutes into the second half. Thomas Vermaelen sent a high ball into the box, where the Frenchman held off both Winston Reid and Andy Carroll to take it down with one sublime left-foot touch before rolling his shot between the goalkeeper's legs.
Arsenal made sure of a crucial victory in the battle for Champions League qualification when Podolski crashed home his second goal of the night with 12 minutes left after a knockdown from Aaron Ramsey.
"The Champions League is very important and we must focus on the next game to ensure that we can qualify," said Poldolski. "It was important to score before half-time and for us to get a win in a local derby.
"The FA Cup is after the season, we leave it now and focus on the Premier League because everyone wants to play in the Champions League.
"The atmosphere is amazing and it is important for the club."