The Gunners took the lead for the first time 10 minutes after the restart when Olivier Giroud produced a moment of magic inside the box, while Podolski grabbed his second of the game with 12 minutes to go to seal the victory.
Here, Sports Mole takes an in-depth look at three important points for Arsene Wenger's side.
On target: 8
On target: 2
Was the result fair?
Yes. Arsenal were once again nowhere near their best, especially in the first half, but they did enough to win the game as West Ham tailed off as time went on. The Hammers arguably had the better of the first 45 minutes and were perhaps unfortunate to be pegged back before the break, but over the whole 90 minutes, Arsenal deserved their win.
It did appear for the best part of an hour as if the Gunners would be on their way to a fifth consecutive league match without a win, but once they took the lead they began to control proceedings and never looked like conceding again.
You could certainly tell that Arsenal had played a gruelling 120-minute semi-final just three days ago. The hosts were sluggish and laboured in their passing and defending, particularly in the first half, and that lack of pace meant that the West Ham defence had little trouble dealing with them.
Podolski's goal just before half time was crucial for Arsene Wenger's side, who hadn't really looked like getting back into the game before it. Had they gone into the break behind, it would have been hard to see them producing as good a second half display as they did.
They were still short of their best after the break, but once Giroud scored his goal they began to keep the ball better and control the tempo of the game. They showed good character to come from behind and win the match, but they can't afford to start that slowly too often if they want to stay in fourth place.
West Ham's performance
It all started so well for West Ham as they carved out the first openings of the game in the early stages, with Mohamed Diame in particular enjoying possession in dangerous areas. They deserved their lead when Jarvis nodded in from close range, but will be disappointed to have not held on to it until at least half time.
The second half was a different story to the first, particularly after Arsenal had taken the lead. The Hammers had always looked capable of causing problems on the break before Giroud's goal, but they never looked like getting back into it after falling behind.
The performance rather fizzled out for the visitors as the match wore on. Whether that was due to the fact that they have little to fight for or because Arsenal simply grew into the game is up for debate, but after a good first-half performance they ended up losing this one with a bit of a whimper.
Sports Mole's man of the match
Santi Cazorla: It is no coincidence that Santi Cazorla became more influential in the game as Arsenal began to control it. The Spaniard ran the show from midfield at times and kept his side ticking over, even in the first half when they weren't at their best.
His assist for the opening goal showed great vision as he slipped the ball through to Podolski to finish, and he also came close to grabbing a memorable strike of his own, fizzing a swerving effort narrowly wide.
This goes to Stewart Downing's attempted clearance at the end of the first half. With the Hammers looking to see out their lead until the break, Downing could only clear the ball as far as Cazorla, who was in a dangerous position.
Sure enough, the Spaniard found Podolski, who in turn found the bottom corner for an important goal that changed the atmosphere inside the ground and the feel of the match. Downing needed to get more distance on his clearance or simply put it out for a throw.
Kevin Friend had quite a quiet game tonight, with no major talking points. Sam Allardyce appeared to be unhappy with a number of decisions from the officials, but they were all relatively minor calls which didn't have a big impact on the game.
Arsenal: Next up for Arsenal is a trip to the KC Stadium to take on Hull City on Sunday.
West Ham: The Hammers, meanwhile, have another London derby as they host Crystal Palace at Upton Park.