The Senegalese forward, signed from Metz in the close-season, netted for a sixth successive game in all competitions shortly after half-time as Sam Allardyce's men came away from Turf Moor with all three points.
That goal continued Sakho's record of scoring in every game he has started this season, and strike partner Enner Valencia doubled the visitors' lead soon after with a superb header from Carl Jenkinson's delivery.
George Boyd reduced the deficit on the hour mark to give Burnley belief, but substitute Carlton Cole nodded home from a corner 20 minutes from time, putting the seal on West Ham's victory.
The result means Burnley - who were on top in the opening half - continue to wait for a first win of the season, and Sean Dyche's men remain rooted in the bottom three with four points from eight matches.
West Ham, meanwhile, have managed back-to-back victories for the first time since March following their 2-0 win over QPR before the international break.
Burnley were boosted by the returns of Danny Ings and David Jones from injury, while West Ham welcomed Mark Noble back into midfield and James Collins came in for the injured James Tomkins.
The home side made a bright start and almost opened the scoring in the fifth minute when Boyd struck the underside of the crossbar with a sweet left-footed half-volley.
Three minutes later Lukas Jutkiewicz saw a close-range finish ruled out for offside, before West Ham's first chance arrived in the 26th minute when Sakho somehow headed wide from four yards after an excellent cross from Aaron Cresswell.
Stewart Downing then cut inside from the right and struck the left-hand upright on the half-hour mark as the visitors began to gain a foothold in proceedings.
Four minutes after the interval, West Ham hit the front through the in-form Sakho.
Once again Cresswell delivered a pinpoint cross from the left, and this time Sakho made no mistake in powering a header into the net from close range.
Five minutes later Valencia doubled West Ham's advantage, meeting Jenkinson's accurate delivery with a thumping header, which left goalkeeper Tom Heaton with no chance.
Valencia was withdrawn soon afterwards, and his replacement Cole saw another headed effort cleared off the line by Kieran Trippier.
That intervention appeared set to prove important as Burnley broke upfield and won a corner that was dropped by Adrian, allowing Boyd to thump home the loose ball.
Ings almost levelled a minute later, but his glancing header from Trippier's fine cross flew narrowly wide with Adrian beaten.
Cole then restored West Ham's two-goal buffer, nodding in from close range after Sakho had headed Noble's corner back across goal.
Burnley substitute Ashley Barnes struck the bar with three minutes remaining to sum up the home side's day as West Ham provided Allardyce with the perfect present on the eve of his 60th birthday.
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