The Eagles were much the better team before the interval but rode their luck after half time before clinching their sixth win of the season.
In contrast Burnley’s winless sequence in all competitions is now seven matches.
After a very quiet opening to the Selhurst Park clash, Wilfried Zaha set up Sako whose first-time effort was blocked.
Palace thought they should have been awarded a penalty when Sako appeared to be fouled by James Tarkowski, but the referee waved away their appeals.
Roy Hodgson’s side deservedly took the lead in the 21st minute though when Christian Benteke set up Sako who fired a deflected shot under Nick Pope in the Clarets goal.
The Eagles almost doubled their lead almost immediately when Ben Mee was forced to clear the ball off the line as James McArthur was about to close in on Zaha’s chip.
As Burnley were forced back Benteke powered a header just over the bar from Luka Milivojevic’s corner.
Just before half time Sean Dyche’s side enjoyed a good spell but Sam Vokes sent his header narrowly wide before Wayne Hennessey saved Ashley Barnes’ dipping volley.
Burnley started the second period well as they went in search of an equaliser but Vokes headed into the side-netting while at the other end Zaha had an effort blocked.
With six minutes left Barnes was denied an equaliser by Hennessey, before Tarkowski’s low cross just evaded Vokes as Palace hung on to claim another three points.