A 3-1 opening day loss at home to Aston Villa resulted in Arsenal supporters questioning Arsene Wenger's status as manager, expressing their unhappiness at the Gunners' lack of progress in the transfer market.
A 3-1 victory in the Champions League qualifying first leg at Fenerbahce quietened the dissenters and first-half strikes from Olivier Giroud and Podolski lifted the mood further.
Germany striker Podolski, who has been linked with a move away from London as Wenger bids to reshape his squad before the September 2 closure of the transfer window, converted another devastating Arsenal counter-attack for a third.
On-loan Aston Villa striker Darren Bent netted on his first Fulham appearance in reply.
Bent has now scored for six clubs in the top flight, but that was as good as it got for the Cottagers, playing in front of new owner Shahid Khan.
Arsenal made two changes from the side which started the win in Istanbul.
Carl Jenkinson replaced Laurent Koscielny, with Bacary Sagna moving to centre-back, and Podolski was brought in for Jack Wilshere, who was on the bench.
Former Tottenham midfielder Scott Parker made his debut for Fulham, who beat Sunderland last weekend, to play for his fifth London club.
Arsenal had the better of the opening skirmishes in torrential rain.
Theo Walcott beat the offside trap to meet Aaron Ramsey's lofted pass with a first-time shot which was saved by David Stockdale, in for shoulder injury victim Maarten Stekelenburg.
Ramsey was getting space and soon afterwards, from the right channel, fired in a shot which hit Giroud, but deflected kindly into the striker's path.
Giroud was composed enough to control the ball before tucking it under Stockdale.
Wojciech Szczesny then made a double save, first stopping an angled effort from the left by Adel Taarabt before quickly getting to his feet to stop the rebound from Damien Duff.
The conditions meant the ball had added zip, while also lacking bounce in some areas, but it was difficult for players to keep their feet.
After Ramsey was booked for one too many mis-timed challenges, Sagna flew in on Dimitar Berbatov, who told referee Howard Webb of his unhappiness at the challenge.
Santi Cazorla hauled down Taarabt, whose influence on proceedings was growing, as the midfielder surged towards the area. Pajtim Kasami's resulting free-kick deflected off the wall for Szczesny to save comfortably.
Ramsey tripped Kasami from behind after 34 minutes and was fortunate to avoid a second yellow card, but Webb gave him the benefit of the doubt.
Fulham had further opportunities down the left, but twice John Arne Riise failed to find his man with crosses.
From the second, Arsenal countered, Walcott burst across the area and found Kieran Gibbs, who shot well wide.
It was a warning for Fulham, who soon afterwards were ripped apart to clinical effect.
Cazorla burst down the centre and found Walcott, whose shot from a narrow angle on the right was parried by Stockdale, but only as far as the edge of the area.
Podolski met the ball and side-footed it first-time into the net.
It could - and possibly should - have been three before the interval.
Having shown his strength, Podolski was indecisive when receiving a pass from Walcott and his return was neither a shot nor a cross.
Fulham were still in contention, but had to convert their chances.
Taarabt had been Fulham's most lively operator in the first half and early in the second his left-foot shot from the edge of the box forced Szczesny to concede a corner.
Ramsey was living dangerously and pulled Taarabt over, but the on-loan QPR playmaker over-hit the resulting free-kick towards the back post.
Bent replaced Duff and Alex Kacaniklic came on for Taarabt, but still Fulham found Arsenal difficult to break down.
Again, direct running at pace was the undoing of Fulham.
Giroud pulled down a long ball for Walcott, who darted up the middle and fed Cazorla.
The Spaniard cut the ball back for Podolski, who just like in the first half, drilled the ball in left-footed past Stockdale.
With 13 minutes remaining, the Cottagers had a lifeline.
Berbatov pulled down a ball over the top and forced Szczesny to save.
Bent was the first to react, tapping in his first Fulham goal.
But it was too little, too late, as Arsenal, who were denied a fourth when Stockdale saved from Walcott, completed victory.