The former Wolverhampton Wanderers forward got the matchwinner, thus, taking his tally to three goals in 15 top-flight games this season.
After dominating the game in the first 20 minutes, Sako handed the Eagles a deserved lead with a left-footed drive in the 23rd minute.
The first half effort stood as the only difference in the encounter as Roy Hodgson's men secured their sixth win in the Premier League this campaign.
Wilfried Zaha who played for the entire duration - his 17th league appearance this season was a thorn in the Clarets' defence but could not find the back of the net.
Crystal Palace moved up to the 12th spot in the Premier League log with 25 points from 23 games, they are now five points above the relegation zone.
They visit Arsenal on January 20 for their next league outing.