Arteta was handed the chance to settle a simply superb match, which lacked any defensive responsibility, but he was denied by an excellent Mark Schwarzer stop that continued Arsenal's frustrating run of form.
After a bright start from Fulham, it was the home side who took the lead on 11 minutes when the excellent Olivier Giroud got ahead of Aaron Hughes and headed home Theo Walcott's corner.
Fulham lost Kieran Richardson to a hamstring injury and, while they were waiting to replace him, Arteta set up Podolski to make it 2-0 12 minutes later, the German getting in front of Sascha Riether to flick past Schwarzer.
But the visitors were never lacking in threat and pulled a goal back on 29 when Dimitar Berbatov was lost by at least four defenders and headed home a Bryan Ruiz corner.
And five minutes before half-time, the Emirates fans were left wondering where their back-four were again when Berbatov ran on to a finely-judged ball down the right and Alex Kacaniklic headed past Vito Mannone, who will think he could have done more to keep the ball out of the net.
Fulham nearly snatched the lead on the stroke of half-time but Ashkan Dejagah couldn't quite get enough on John Arne Riise's cross and Arsenal breathed a sigh of relief.
Ruiz snatched at a decent chance just after the start of the second-half and, at the other end, Walcott cut in from the flank and fired just over the bar from 20 yards.
More dreadful defending, this time from Arteta, gave Fulham the lead on 67, the Spaniard losing possession and bringing down Ruiz in the box, with Berbatov making no mistake from the spot.
Two minutes later the rollercoaster was back on an even keel with Giroud's shot hitting the post and rebounding for Walcott to cross for the Frenchman to score his second of the afternoon.
Giroud was only denied his hat-trick by a superb save from Schwarzer, turning his fierce 20-yarder around the post, as a topsy-turvy game headed for the final 10 minutes.
And there would be more drama in the closing seconds with the last kick of the game proving to be Arteta's penalty, given for a marginal handball, but Schwarzer came to the rescue with a superb save to give Fulham a well-deserved point.