On what would have been Bill Shankly's 99th birthday, the Reds equaled the record of his newly promoted side in 1962-63 by collecting just one point from their opening three games.
After a lack of activity on deadline day, both owner John Henry and manager Brendan Rodgers are now subject to criticism.
Gunners manager Arsene Wenger will reflect that he could easily have been in a similar 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 halftime, the Gunners chief can start looking forward with confidence.
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.
The impression Liverpool were in control was a misleading one. With Cazorla continuing to impress and Abou Diaby thriving in space, 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.
However, after 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.
Liverpool emerged with more purpose after the break, and once Stewart Downing was introduced they had more penetration. The winger had a shot deflected wide by Vermaelen not long after Per Mertesacker, who had been booked very early on, had sent Luis 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.
Substitute 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.
At the other end, Vito Mannone twice denied Shelvey and Suarez lifted his shot over after turning onto Gerrard's pass.