Germany international Podolski equalised before the break and capped a fine display with the third goal, after Olivier Giroud had put Arsenal in front 10 minutes into the second half, as Arsene Wenger's side worked hard to claim the three points.
The result takes Arsenal up a spot to fourth on 67 points with four games left, one place and one point ahead of Everton who have a game in hand against mid-table Crystal Palace today as the Champions League qualification race hots up.
West Ham's Matt Jarvis gave the resolute visitors a deserved 40th minute lead over a tired-looking Arsenal, who were taken to penalties by FA Cup holders Wigan Athletic in the semi-finals on Saturday, when he nodded home a loose ball in the area.
The home side's response was swift, though, and they levelled in the 44th when Podolski found space in the Hammers box and his precise, driven finish beat goalkeeper Adrian.
Frenchman Giroud, who has struggled to find his best form in recent weeks, made it 2-1 when he controlled the ball sublimely with his left foot before striking a low shot through Adrian's legs with his right after 55 minutes.
Podolski then blasted in his second after being set up by Aaron Ramsey's precise header from a Giroud cross to seal the win in the 78th minute much to the relief of the home fans.
"It was a hard-fought win and a very enjoyable win as we had to dig deep on Saturday," Wenger told Sky Sports. ""We were under absolute pressure to win the game, on top of that it was a derby and, after giving a lot on Saturday, it was difficult.
"Maybe we could have scored more, I was satisfied with the mental response of the team tonight, the focus tonight of the team was outstanding," added the Frenchman.
Having led the Premier League for much of the season, Arsenal's capitulation since late January has been so drastic they have gone from title contenders to a side struggling to earn a place in Europe's elite competition.
Before the visit of West Ham they had picked up eight points from 21 while the form of Everton, who have won seven straight games including a 3-0 win over Arsenal earlier this month, means the Gunners face a real scrap to secure qualification.
The win ended a run of four league games without a victory for Arsenal and Podolski immediately turned his attention to the next league game at FA Cup final opponents Hull City on Sunday as they aim for for a 17th straight Champions League campaign.
"The Champions League is very important and we must focus on the next game to ensure that we can qualify," he told Sky Sports. "It was important to score before halftime and for us to get a win in a local derby."
"The FA Cup is after the season, we leave it now and we 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," Podolski added.