Podolski scored twice after his forward partner had grabbed the opener for the visitors, who put to bed all talk of an early season crisis with their first league points of the season.
Liverpool secured a second win from their opening two games and went level on points with Chelsea at the top of the table as they beat Aston Villa 1-0 thanks to a superb individual goal from Daniel Sturridge.
Everton drew their second successive game after failing to break down West Bromwich Albion in a 0-0 draw at Goodison Park, while Newcastle United's match at home to West Ham United was also scoreless.
It is the second time in four days Arsenal have scored three goals, following Wednesday's 3-0 victory over Fenerbahce in their Champions League playoff.
Saturday's win and the manner of the performance will go some way to making up for their opening day slip-up at home to Aston Villa, while the names on the scoresheet might make fans think twice about demands for forward reinforcements.
"Yes, I'm pleased because we played like a real team. It is never easy to play away in the league and we did it convincingly," Arsenal manager Arsene Wenger told BT Sport.
Despite their resounding midweek win, the pre-match talk still focused on Arsenal's need for reinforcements and the club's perceived failure to land a signing to excite frustrated fans.
"You can get people to focus too much on who you will buy and forget how good the players are here," Wenger added.
Giroud, who is often touted as one of the men Arsenal need to upgrade if they are to chisel out a title challenge, had his say in the debate with his third goal in as many games this season on nine minutes.
An Aaron Ramsey shot fell neatly into the Frenchman's path and he lifted it delicately over keeper David Stockdale and into the net.
In atrocious conditions that made Arsenal's intricate passing game difficult, the visitors doubled their advantage four minutes before the break.
Podolski followed up a shot from Theo Walcott that was parried and blasted his effort into the net.
The German made sure of the points midway through the second half when he fired in from the edge of the area for his second, before Darren Bent grabbed a consolation with a poacher's finish on his debut.
Liverpool, who have also come in for criticism from fans for failing to pull off a big-name summer signing, took the lead at Villa Park through Sturridge after 21 minutes.
The England striker controlled a neat dummy from Philippe Coutinho, shifted the ball out of his feet and away from two defenders before rounding Villa keeper Brad Guzan and stabbing high into the net for his second goal in as many games.
After his penalty-saving heroics against Stoke last week, Liverpool keeper Simon Mignolet produced two excellent stops to deny Christian Benteke, the second of which was an acrobatic one-handed push around the post.
Everton found West Brom keeper Ben Foster in inspired form, despite being forced off injured with what his manager called a 'serious' foot injury while West Ham boss Sam Allardyce returned to former club Newcastle and frustrated his old employers.
Hull City supporters staged a protest against a proposed change of the club's name to Hull City Tigers shortly before kickoff, but the fans would have been satisfied with the result.
Yannick Sagbo was brought down in the box and Robbie Brady stepped up find the net from the spot on 22 minutes. Sagbo was then sent off five minutes later for a headbutt, but Steve Bruce's side held on.
Charlie Adam, facing former boss Ian Holloway, who coached him at Blackpool, equalised on 58 minutes with a precise finish at the far post and Ryan Shawcross put them ahead with a powerful shot from eight metres.
Giaccherini scored his first Premier League goal when he got on the end of a Seb Larsson corner to head Sunderland into a third-minute lead against Southampton, but Fonte equalised with a glancing header.