Former Wolves forward Sako was aghast to see a first-half penalty claim waved away after being brought down by James Tarkowski but channelled his frustrations to secure the points midway through the first half at Selhurst Park.
Palace's recovery under Hodgson now stands at a solitary defeat in its past 12 Premier League outings – an impressive rehabilitative effort for both the south London club - five points clear of the bottom three.
That poisoned chalice might one day pass to Sean Dyche, who has rarely seen his robust backline endure as many problems as Sako, Christian Benteke and Wilfried Zaha inflicted during the opening half hour.
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.
Burnley left-back Charlie Taylor cleared a viciously dipping Patrick van Aanholt cross just over his own crossbar in the eighth minute – a close shave that encapsulated a nervy opening for the visitor.
Palace should have been awarded a penalty when Tarkowski dived in desperately on a rampaging Sako, although the same player responded by superbly breaking the deadlock in the 21st minute.
There was still plenty to do when Sako retrieved Benteke's flick, but he drove towards a backpedalling Phil Bardsley and crashed a low shot through Tarkowski's legs that flew into the net via the boot of Nick Pope.
Pope was then sold short by Taylor, with Zaha beating the Burnley goalkeeper to an ill-advised header back. Captain Ben Mee flung himself bravely to brilliantly deny James McArthur on the goal-line.
Benteke leapt to power over from a Luka Milivojevic corner, while a set-piece situation saw Dyche's side belatedly threaten – Vokes glancing Johann Gudmundsson's searching ball from the left just past the far top corner.
Palace keeper Wayne Hennessey did well to tip Barnes' dipping effort over, while Pope would have had no chance in the 44th minute had Sako's blistering volley from Timothy Fosu-Mensah's cross been on target.
Fosu-Mensah was less effective inside his own area six minutes into the second half when Barnes got over the back of him to nod wide, while Vokes came closer still by heading into the side netting.
A bouncing ball in the Burnley box caused confusion in the 57th minute and Taylor did well to block when the goal beckoned for Zaha.
Georges-Kevin Nkoudou was introduced in place of Jeff Hendrick for a Burnley debut after the hour and the Tottenham loanee almost had an immediate impact, clipping in a cross from the byline on the left that Gudmundsson was unable to turn home.
Benteke headed off target under pressure from the tireless Taylor when Sako stood up a 69th-minute cross to the far post - more fine combination play from a forward duo who might relish next weekend's trip to Arsenal.
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.