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.
Here are the player ratings.
1. Nick Pope 5.5
Aside from the goals, which he had very little hope of stopping, the England international was only required to make a one-handed save to deny Saint-Maximin in the first half. However, his kicking was poor throughout, which handed possession to the opposition as he put many clearances into touch.
Photo: Ryan Pierse
2. Connor Roberts 4
Sacrificed at the break as the Clarets introduced Weghorst in search of an equaliser. Asked to get higher up the pitch when the Clarets had possession, which often afforded Saint-Maximin the space to run into when the ball was turned over. Wasted a couple of opportunities to play Cornet in.
Photo: Alex Livesey
3. Charlie Taylor 5
Had his passage down the left hand side restricted by Trippier and a number of supporting midfielders, though managed to land the ball on Cornet's head with a peach of a cross at 2-1. However, similarly to when he lost Watkins against Villa earlier in the month, the defender afforded Wilson the time and space to tap home from close range when caught on his heels.
Photo: Richard Heathcote
4. Nathan Collins 4
The Republic of Ireland international was inconsolable at full-time when relegation was confirmed. The centre back had a nightmare couple of minutes when a couple of errors gifted Newcastle the lead. He conceded the corner after allowing Bruno to have a shot on goal and then handled the resulting set-piece. Failed to cut out Saint-Maximin's cross for Wilson's second.
Photo: Jan Kruger