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.
Palace were always a threat on the counter and fashioned the first chance as a Clarets quickly turned into a break for Ian Holloway's side which ended with Glenn Murray firing 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.
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.
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 Dean Marney headed straight at Julian Speroni from Trippier's cross.
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 Mile Jedinak hit the bar in stoppage time, but Sean Dyche's side held on for the three points.