Germany international Podolski scored one in each half after Olivier Giroud had opened the scoring early on, before Darren Bent grabbed a late consolation for Martin Jol's men after coming off the bench for his home debut.
Fulham boss Jol made three changes from his side's 1-0 win at Sunderland, handing a debut to new signing Scott Parker and also bringing in David Stockdale in goal and John Arne Riise at left-back.
There was an element of good fortune about the goal as Aaron Ramsey scuffed a shot from 20 yards that caught the striker's heel and ran perfectly into his path.
The hosts continued to press for an equaliser but were served a warning of Arsenal's potency on the break when Walcott sprinted clear down the right and teed up Kieran Gibbs, only for the full-back to drag his shot wide.
Stockdale saved the England forward's shot well after he had been played in by Cazorla, but Podolski followed up with an unerring side-footed finish from 18 yards to send Arsenal in two goals to the good at the break.
Fulham flew out of the traps with real intent in the second half and Szczesny saved smartly from Taarabt's drive just two minutes after the restart.
But Arsenal were again dangerous on the break, and Ramsey almost added a third in the 57th minute after Walcott had nodded Gibbs' deep cross back to the Welshman, but his shot was deflected wide.
That proved to be a pre-cursor for the third goal, as another slick counter-attack led to Wenger's side increasing their advantage on 68 minutes.
Substitute Bent gave the hosts a glimmer of hope nine minutes later with a typical poacher's finish after Szczesny had parried Dimitar Berbatov's clever half-volley, but Arsenal held on to complete their sixth consecutive away Premier League victory.
