Forwards Sam Vokes and Ashley Barnes had the Clarets' best opportunities to equalise but a slump from their sparkling form earlier in the season now sees them winless in the past six Premier League matches, with Manchester United up next at Turf Moor.
But the hosts were soon ahead, through the Malian global, on 21 minutes.
In the 21st minute, Wilfried Zaha's aimless pass was flicked on by Christian Benteke to Sako on the left wing and he eased into the area before shooting inside the far right post via goalkeeper Nick Pope's foot.
They were fortunate to escape further punishment when Charlie Taylor's weak header was intercepted by Zaha and it required a brilliant last-ditch clearance by Ben Mee to prevent James McArthur from converting.
Benteke had a chance and Sako went closer still on the stroke of half-time.
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Palace continued to dominate their opponents as Benteke nodded over from a Luka Milivojevic corner, while Sako volleyed just wide from a deep cross from Timothy Fosu-Mensah. In the first half we created enough chances to have made things more comfortable for ourselves in the second half.
It was Sako's sixth goal of the season.
Hennessey was out sharply to deny Barnes an equaliser four minutes from time, with a subsequent melee by the touchline after Taylor sent Zaha careering into television equipment indicative of Burnley's frustration.
The Red Devils are now overstocked in the positions Fosu-Mensah would normally be considered to play in and so he was sent out on loan to another Premier League club to learn his trade.
Schlupp, 25, has been a regular fixture for Crystal Palace having played in 19 of their 22 English Premier League fixtures this season.
Crystal Palace are playing well and with the form they are in, they will hope to avoid relegation easily.