Licka was without a club when Burley signed him in the summer after being released by Czech side FC Slovak.
The blond bombshell thanked Burley for taking a chance on him by scoring his first goal on English soil in the 54th minute, to end Stoke's mini revival.
Licka cut in from the left before hammering the ball past Stoke goalkeeper Steve Simonsen's despairing dive and into the bottom left-hand corner from 20 yards out.
Saints' top-scorer Grzegorz Rasiak should have opened the scoring in only the sixth minute when he flashed a free header over the crossbar after Andrew Surman had crossed from the left.
Inigo Idiakez was denied a stunning opener by a superb piece of goalkeeping from Simonsen nine minutes later. Idiakez unleashed a curling 20-yard pile-driver, which was heading for the top left-hand corner until Simonsen came flying across his goal to turn it over for a corner.
Southampton old boy Ricardo Fuller brought a smile to the home fans' faces soon after when he skied a left-footed shot from the edge of the penalty area.
Home keeper Kelvin Davis matched Simonsen's heroics midway through the half when he pulled off a fine fingertip save to push Lee Hendrie's fizzing right-footed shot around his right-hand post.
Fuller was given another chance to mark his return to the St Mary's Stadium with a goal three minutes before half-time when he rose to flick Hendrie's left-wing centre goalwards, but it went straight into the arms of a grateful Davis.
Licka lit up a numbingly dull game with his delicious goal eight minutes after the re-start.
Hendrie threatened an instant response from the visitors when he fired a 20-yard free kick narrowly wide of Davis's right upright.
Simonsen then came racing out of his goal to deny Saints a second goal 13 minutes from time, just managing to beat the on-rushing Kenwyne Jones who was aiming to get on the end of Idiakez's throughball.
Surman almost put the perfect seal on the victory with a sweet 25-yard lob that landed on the roof of the Stoke net as Saints cranked up the pressure in the latter stages to end any hopes of a fightback from the visitors.