Following a tentative start by both sides, the Mali international was denied the opportunity to score from the spot after James Tarkowski brought the winger down, but referee Michael Oliver waved away the appeals.
On 21 minutes, however, the game's first effort on target yielded the first goal as Sako cut in from the left after good work by Christian Benteke before rifling the ball between the legs of Clarets goalkeeper Nick Pope.
Palace were comfortable for most of the game, save for a late ball over the top which found Barnes and saw Wayne Hennessey rushing out of his box to make a fine save.
The result sees Palace edge up to 12th in the table, while Burnley remain seventh, five points behind Arsenal.
CRYSTAL PALACE (4-5-1): Hennessey; Fosu-Mensah, Kelly, Tomkins, Van Aanholt; Zaha, Milivojevic, Riedewald, McArthur, Sako; Benteke.
BURNLEY (4-2-3-1): Pope; Bardsley, Tarkowski, Mee, Taylor; Cork, Defour; Gudmundsson (Wells 85'), Hendrick (N'Koudou 63'), Barnes; Vokes.