The Clarets finished three points adrift of Leeds United, who beat nine-man Brentford with the last kick of the game to preserve their status in the top flight.
Burnley were downed by a Callum Wilson double; the first from the spot after Nathan Collins’ needless handball from a Kieran Trippier corner.
The striker then doubled the Magpies’ advantage when guiding Allan Saint-Maximin’s cross beyond Nick Pope from close range.
Leading scorer Maxwel Cornet thumped the ball past Martin Dubravka on the volley to give Mike Jackson’s men a lifeline.
They pressed for an equaliser, but substitute Wout Weghorst poked the ball wide from close range, Jack Cork’s header was cleared off the line and a corner header was saved by Dubravka.
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The striker then doubled the Magpies’ advantage when guiding Allan Saint-Maximin’s cross beyond Nick Pope from close range.