Sanchez drilled home Aaron Ramsey's pass in the 88th minute following a frantic passage of play with the Gunners taking full advantage of Saints having been reduced to 10 men.
When Toby Alderweireld went off injured in the last 10 minutes, Southampton had already made all three permitted substitutions and as they became stretched Arsenal pounced.
Few chances fell at either end in an uninspiring first half that did little to alleviate the chill in north London with Saints striker Graziano Pelle guilty of one glaring miss from close range.
Southampton were livelier after the interval, attacking with greater intent than their strangely subdued opponents, whose quest to narrow the gap on the top four lacked urgency on a night of high frustration for home supporters.
The 65th-minute arrival of Olivier Giroud lifted Arsenal, although the substitution was not without controversy as Arsene Wenger's decision to sacrifice the lively Alex Oxlade-Chamberlain was met with boos.
Santi Cazorla had endured an error-prone night, but it was Oxlade-Chamberlain who made way in a performance that will only provide further ammunition to critics who believe the club is stagnating under Wenger.
Giroud was denied with three minutes to go after Southampton keeper Fraser Foster palmed his header to safety, but seconds later Sanchez plundered the winner as Saints cracked under the pressure.
Arsenal had the first chance as a perfectly-weighted pass from Cazorla sent Danny Welbeck goalbound but the England marksman could only find the side-netting from a tight angle.
Southampton were then left wondering how they had failed to take the lead when Pelle blasted a close-range shot over the crossbar after combining with Steven Davis on the edge of the area.
Arsenal were playing with fire as Nacho Monreal and Mathieu Flamini combined to set up a panicky back-pass that left Emiliano Martinez with little margin for error as he cleared.
It was then the turn of Saints to panic as Cazorla opened up their defence only for Oxlade-Chamberlain to direct his shot straight at Forster.
A mix-up at the near post created the narrowest of openings for Welbeck but his backheel struck the side-netting one more.
Forster plucked a floating free-kick by Cazorla out of the air as Arsenal probed once more against opponents they were finding difficult to break down.
Cazorla, Sanchez and Oxlade-chamberlain dazzled on approach into the area, but the final pass was repeatedly blocked, while at the other end Pelle was Saints' main troublemaker.
The strength of Victor Wanyama dispossessed Sanchez on the halfway line and the Kenyan midfielder had the wits to find Pelle, who tested Forster from 25 yards out.
Only Forster's fingertip save of a deflected header by Welbeck kept the ball out of the net as play swept to the other end.
Southampton made the brighter start to the second half as Pelle saw a shot blocked by Laurent Koscielny before Shane Long butchered a great chance after Arsenal had opened up on the counter.
Giroud, who had been replaced by Oxlade-Chamberlain in the starting line-up, stepped off the bench as Wenger sought to ignite his team and among the Frenchman's first touches was one to force a save by Forster.
Arsenal now had discovered some energy and they fashioned another chance with Sanchez picking out Welbeck only for the alert Forster to intervene once more.
The draw looked to be within Saints' grasp, only for Alderweireld's exit to leave them exposed with Sanchez landing the killer blow.
Source : PA