After securing just five points from as many Barclays Premier League matches, the Gunners were in danger of losing ground on the top four coming into the game.
Arsene Wenger's side looked set for another frustrating day at the office on Saturday - thanks in no small part to an outstanding display by Stoke goalkeeper Asmir Begovic.
The 25-year-old proved why he is so sought-after with a fine first-half display in which he superbly denied Alex Oxlade-Chamberlain either side of stopping a Laurent Koscielny header.
It was those in front of Begovic that came to the fore in a second half in which it looked like City's stubborn defence would help end a run of 10 successive defeats in league and cup at Arsenal.
However, substitute Podolski ended those hopes 12 minutes from time with a low free-kick that deflected in off Geoff Cameron.
The Gunners made five changes for Saturday afternoon's match, with deadline-day signing Nacho Monreal handed his debut.
The Spain international replaced the injured Kieran Gibbs at left-back in an Arsenal side that started much better than they had against Liverpool in midweek.
Robert Huth blocked early threatening balls from Theo Walcott and Jack Wilshere, before Oxlade-Chamberlain, making his first league start since January 13, struck wide.
Wilshere blazed over after a Walcott cutback failed to find a team-mate as Arsenal continued to press for an early goal.
It took 15 minutes for Wojciech Szczesny to best tested by Stoke.
Jonathan Walters' strike from distance was comfortably dealt with by the Poland international, who then watched a Peter Crouch snapshot deflect off Koscielny and out for a corner.
There were rare moments of relief for a Stoke side on the backfoot, with Olivier Giroud heading a Abou Diaby cross into the ground when Arsenal returned to the attack.
Andy Wilkinson then blocked a right-footed strike from Walcott and, after Ryan Shawcross sent over a wayward volley, City goalkeeper Begovic came to the fore.
First the Bosnia international denied Oxlade-Chamberlain from close range after a Wilshere corner fell straight into his path.
Begovic was the scourge of Arsenal once again in the 34th minute, denying a Koscielny header with another block.
Four minutes from the break the Stoke goalkeeper was at it again.
Wilshere played an exquisite through ball to Oxlade-Chamberlain, whose curled effort looked destined to ripple the net until Begovic superbly palmed wide.
Arsenal continued to press for the opener as half-time approached, but the only action of note was a clash of heads between Monreal and Walters.
The latter got a bloody head in the challenge, but was patched up with a bandage similar to the one donned by team-mate Huth.
The German centre-back was called into action shortly after play resumed for the second half, rising well to stop Giroud bundling home a Monreal cross.
Giroud fired wide under pressure soon after but Arsenal's attacks were becoming less frequent.
Wenger brought on Santi Cazorla and Podolski in a bid to inject new life into his side's attack, with Diaby and Oxlade-Chamberlain replaced.
The introduction of the Spaniard almost had an immediate impact as Cazorla sent in a floated cross to Giroud, but the Frenchman was unable to direct his header on target.
Glenn Whelan picked up a booking as City looked to hold off the hosts, who finally took the lead in the 78th minute.
Wilkinson received a yellow card for bringing down Walcott on the edge of the box, allowing substitute Podolski to send a low free-kick that deflected in off Cameron's heel.
It had looked like the goal would be chalked off for an apparent offside against Walcott, but, after consulting with his assistant, referee Chris Foy awarded the goal.
Cazorla came close to making it 2-0 shortly afterwards, only for Begovic to get the slightest touch to send it behind.
Michael Owen, Kenwyne Jones and Cameron Jerome all came on for the final stages at Tony Pulis looked for a leveller, which never came in a heated end to the game.
Shawcross was booked for a late challenge on Koscielny and Owen looked to aim a punch at Mikel Arteta after a late challenge by the Spaniard.