On what would have been Bill Shankly's 99th birthday, the Reds equalled the record of his newly-promoted side in 1962-63 by collecting just one point from their opening three games.
Coming on the back of Friday's deadline-day transfer debacle, when they failed to secure a replacement for Andy Carroll who had joined West Ham 24 hours earlier, the situation has provided huge cause for concern on Merseyside, with both owner John Henry and manager Brendan Rodgers in the firing line.
Arsene Wenger will reflect that he could easily have been in the same position had Arsenal lost.
However, with Cazorla capping an excellent display by profiting from a Jose Reina blunder after summer recruit Podolski had opened their account for the season before half-time, the Gunners chief can start looking forward with confidence.
A summer of optimism has given way to a sobering reality at Anfield.
Henry did spend big in his backing of Kenny Dalglish but as only Luis Suarez of those expensive purchases started against Arsenal, the wisdom of Dalglish's work has to be questioned.
So much now rests on Suarez and for the first half at least, he almost seemed to be trying too hard.
Fabio Borini and Raheem Sterling had Liverpool's best chances before the break.
The latter continues to catch the eye and his excellent turn onto Steven Gerrard's knockdown created the half-chance he belted against the outside of a post.
Borini was rather more rash with his openings, two of which he sent sailing over the bar.
Worst of all, Daniel Agger failed to head home a Gerrard corner when he had been left completely unmarked.
The impression Liverpool were in control was a misleading one.
With Cazorla continuing to impress and Abou Diaby thriving in the space Nuri Sahin failed to fill, Arsenal looked capable of opening their opponents up on the counter.
That worrying statistic of potentially becoming the first side to start the season with three successive scoreless games hung heavy for a while.
But after Cazorla's shot had been saved by Reina, the Spain midfielder was instrumental in Arsenal breaking their duck for the campaign.
A wayward Gerrard pass allowed Thomas Vermaelen to set up a break from the edge of his own box. Podolski fed Cazorla, then kept motoring. By the time Cazorla was in a position to play the killer pass, Podolski had arrived in support and the German drilled expertly past Reina from 10 yards.
The visitors should have doubled their advantage almost immediately.
Diaby's surging run out of his own half was reminiscent of Patrick Vieira in his pomp. He released Olivier Giroud inside the Liverpool box, but the finish was poor and Reina watched it fly wide.
Liverpool emerged with more purpose after the break, and once Stewart Downing was introduced they had more penetration too.
The winger had a shot deflected wide by Vermaelen not long after Per Mertesacker, who had been booked very early on, had sent Suarez to the deck with a clumsy challenge that could easily have brought Liverpool a penalty.
As it turned out, Arsenal were merely softening their opponents up for another body blow.
Just on in place of the ineffective Sahin, Jonjo Shelvey failed to track Cazorla's arcing run around the corner of the Liverpool area.
After an excellent one-two with Podolski, Cazorla also left Downing standing before firing a shot goalwards from an acute angle which Reina was unable to keep out.
The goal merely heightened growing worries about Reina, whose form has dipped below the high standards he set a couple of years ago.
At the other end, Vito Mannone twice denied Shelvey and Suarez lifted his shot over after turning onto Gerrard's pass.
"You're getting sacked in the morning," taunted the visiting supporters to Rodgers before the final whistle.
He isn't. But with the squad at his disposal, the Northern Irishman will do well to preside over any substantial improvement on last term's eighth-placed finish.