Szczesny was caught badly out of position after racing out of his goal, allowing Sadio Mane to conjure a spectacular finish and give his side a 34th-minute lead at St Mary’s.
Then 11 minutes into the second half, the Poland international’s poor clearance allowed Dusan Tadic to net the decisive second.
The victory extended Southampton’s improved run of form after they suffered a run of five successive defeats, although it came at the cost of an injury to Senegal winger Mane that could threaten his involvement in the Africa Cup of Nations.
But for Arsenal, defeat denied them the opportunity to leap-frog their opponents and left them three points adrift of the Saints in the race to qualify for the Champions League.
Southampton have emerged as surprise contenders for a European place and their gritty reaction to their recent run of poor form suggests they are in it for the long haul.
Saints manager Ronald Koeman added to the rivalry between these two teams by insisting beforehand that his club could profit from Arsenal’s crowded schedule when the Champions League resumes in February.
A shortage of fit midfielders meant Wenger was forced to pair Francis Coquelin, recalled recently from a loan spell at Charlton Athletic, with centre-back Calum Chambers in front of the back four, while Alexis Sanchez started up front in the absence of the suspended Olivier Giroud and injured Danny Welbeck.
There was certainly a disjointed look to the visitors as Koeman’s side quickly pressed, exposing Arsenal’s right flank twice in the opening minutes.
Graziano Pelle should have headed the home side into a sixth-minute lead when he struck the post from Tadic’s hooked cross.
Then two minutes later Szczesny was forced to scramble Mane’s shot away after his defence had been caught flat-footed.
– Rush of blood –
Arsenal struggled to find their usual fluency but Alex Oxlade-Chamberlain fired narrowly over from a similar opportunity and it appeared as though Arsenal were beginning to settle when Szczesny’s rush of blood gifted Southampton the lead.
The ‘keeper found himself stranded on the left-hand edge of his penalty area when he attempted to prevent Mane chasing onto a loose ball, and with Szczesny stranded, the forward curled in a left-foot shot from the tightest of angles.
Wenger was clearly incensed by the way his side had conceded but he had cause to thank his ‘keeper five minutes before the break when the Pole saved well with his legs to deny James Ward-Prowse.
Arsenal needed to find a reaction after the break if they were to force their way back into the match and Sanchez drew an excellent save from Fraser Forster 10 minutes after the restart.
But less than a minute later, Szczesny was again at fault as he gifted Tadic the chance to open up a two-goal lead.
The visitors were again unsettled by a long ball that allowed Steven Davis to get in behind the defence and drill in a low cross.
With no Saints players on hand to turn the ball home, it appeared as though Mathieu Debuchy had snuffed out the danger when he blocked the cross inside the six-yard box, but the ball rolled towards the grounded Szczesny who hacked it straight to Tadic.
The Serb then had the straightforward task of stabbing home from close range and Arsenal were in deep trouble.
The situation could have got worse for Wenger’s side with Pelle again hitting the woodwork before later seeing his shot headed off the line by Debuchy.