Arsene Wenger's side always carried a menacing edge to their play as efforts from Olivier Giroud (14) and Podolski (41, 68) completed a comfortable win with substitute Darren Bent (77) finding a consolation for the home team.
Podolski added the second by converting from 12 yards after Stockdale had blocked a powerful Theo Walcott hit before weighing in with a third from a Santi Cazorla pass. Bent scored from close range after Wojciech Szczesny could not hold a Dimitar Berbatov cross, but the visitors were good value for three points.
Wenger will be content with the afternoon's work coming on the back of Wednesday's 3-0 away win in the Champions League play-off first leg against Fenerbahce in Turkey.
PRE-MATCH: PARKER FULHAM DEBUT, JENKINSON STARTS FOR VISITORS: Scott Parker made his debut for Fulham in midfield while goalkeeper Maarten Stekelenburg missed out due to injury. Darren Bent was on the bench. Carl Jenkinson came in for the suspended Laurent Koscielny for Arsenal, with Bacary Sagna moving to centre-back, and Podolski was in for Jack Wilshere, who was rested.
14' - ARSENAL MOVE AHEAD: Ramsey's shot from distance breaks kindly for Giroud. One touch takes him on goal, and the second touch is chipped over Stockdale. Arsenal lead 1-0.
18' - FULHAM RESPONSE: Adel Taarabt sees a shot blocked by Wojciech Szczesny before the Arsenal keeper is alert to halt Duff's rebound. Smart piece of goalkeeeping.
39' - ARSENAL BREAK: Brilliant counter attack from Arsenal after Fulham had a chance with Riise breaking into the area. Gibbs drags the shot wide of goal. Was a golden chance for a second goal. Fulham living on the edge.
41' - A SECOND ARSENAL GOAL: Podolski slams the rebound into the net after Stockdale had saved from Walcott. Arsenal cutting Fulham apart in the middle of the park.
68' - A SECOND FOR PODOLSKI: Podolski drives the ball low and hard into the corner of the net. What a fine finish from Santi Cazorla's cutback. Brilliant move and finish. Arsenal move three to the good.
77' - BENT MAKES HIS MARK: Arsenal go to sleep. Berbatov turns to knock the ball at goal. Szczesny with the save, but Bent converts from close range. 3-1 win.
88' - ALMOST A FOURTH ARSENAL GOAL: A few of the home fans begin to head for the exit doors as Stockdale saves from Walcott after being sent in on goal by Cazorla.
TALKING POINT: Arsene Wenger gave an increasingly rare start to Podolski and his two goals decided the game - but does he have a long-term future at Arsenal? He is reportedly a target for Schalke and although he offers a clear goal threat, it remains doubtful whether his all-round game brings enough to the Arsenal side. Will Wenger cut him loose before the window closes?
Fulham have lost their last five Premier League home matches, their worst run in the top flight between November 1961 and February 1962, when they were on a club record run of seven consecutive home defeats. Arsenal have won their last six Premier League away matches – their best run for 11 years, since January-May 2002 (the Premier League club record 8 away wins).
Darren Bent extends his record of scoring against Arsenal in the Premier League for the most number of different sides (now 5). 11 players have scored a Premier League goal against Arsenal for three different clubs.
Arsenal have lost just one of the nine previous Premier League matches in which Giroud has scored (six wins, two draws) – the 3-1 home defeat v Aston Villa last Saturday.
MAN OF THE MATCH
Aaron Ramsey (Arsenal) - A delightful performance at the heart of the midfield for the visitors. Kept it simple, and was such a key figure on a tough old day in the rain.
Fulham: Stockdale 6, Riether 7, Hughes 6, Hangeland 6, Riise 6, Duff 5, Parker 7, Sidwell 5, Taarabt 7, Kasami 5, Berbatov 5; Subs: Kacaniklic 6, Karagounis 6.
Arsenal: Szczesny 7, Jenkinson 7, Sagna 7, Mertesacker 7, Gibbs 7, Ramsey 8, Rosicky 6, Cazorla 6, Podolski 8, Walcott 7, Giroud 8; Subs: Monreal 6, Wilshere 6, Sanogo 6.
THE MATCH IN A TWEET
It's Lukas Pogoalski as two thundering finishes help sink Fulham; River Thames spills into sodden Craven Cottage