After the hosts had wasted a host of chances, Stanislas curled a brilliant strike into the top corner from outside the box with nine minutes to go to secure the three points.
Burnley missed Charlie Austin as they squandered a number of first-half opportunities. Palace were always a threat on the counter, though, and fashioned the first chance as Glenn Murray fired narrowly wide.
Burnley had their first sight of goal with 10 minutes gone but Sam Vokes fired over after being found by Kieran Trippier's long ball. Two minutes later Stanislas found Danny Ings in the area but his shot flew wide.
Trippier was a threat down the right and after 24 minutes he fired in from the right and Vokes was inches away from getting a telling touch in the middle.
In-demand winger Wilfried Zaha's first real involvement in the game brought a good save out of Lee Grant after the winger cut in from the left and let fly from the edge of the area.
Dean Marney then created another chance for the Clarets as he got to the byline and pulled back for Ings, but he failed to get a proper touch on the ball and Julian Speroni comfortably saved.
But if it were not for Grant Burnley would have fallen behind as the goalkeeper got down brilliantly to turn the ball away after Jason Shackell had diverted Jonathan Parr's cross towards his own goal.
In the second half Vokes was brilliantly found by Danny Lafferty but he failed to hit the target from close range, while Marney headed straight at Speroni from Trippier's cross.
Zaha was causing problems and won a free-kick on the left with 20 minutes left. After Mile Jedinak's effort hit the wall, Zaha created space for himself but his shot was palmed away by Grant.
But just as it seemed the visitors would go on to claim all three points, Stanislas - who was about to be replaced - picked out the top corner from the edge of the box with his last touch of the game.
Palace pressed for an equaliser and Jedinak hit the bar in stoppage time, but Sean Dyche's side held on for the three points.