Mike Jackson named an unchanged starting line-up from Thursday’s draw at Aston Villa, for the visit of the Magpies.
Callum Wilson’s penalty inside the opening 20 minutes following Nathan Collins’ handball separated the teams at half-time.
Wilson doubled Newcastle’s advantage on the hour with a composed finish in the box.
Maxwel Cornet scored a 69th minute consolation for the hosts, which could not prevent relegation.
First-half
Roared on by a sell-out crowd, the early minutes saw both sides battle to create the first chance.
The game was paused on 11 minutes for Newcastle forward Joelinton to receive treatment for an injury which forced the Brazilian to be stretchered off and replaced by Jacob Murphy.
Nick Pope was called into action to parry Bruno Guimaraes’ strike from the edge of the box, with Collins clearing the rebound before Wilson could reach it.
From the resulting corner, VAR adjudged Nathan Collins to handle in the box, as referee Craig Pawson pointed to the spot.
Wilson was tasked to take the penalty, and he converted his right-footed kick into the bottom right-corner, sending Pope the wrong way to give Eddie Howe’s men a 20th minute lead.
The Clarets pushed for an equaliser midway through the half. Connor Roberts’ cross aimed for Ashley Barnes was dealt with by visiting skipper Jamaal Lascelles while Dwight McNeil struck a rising effort just over Martin Dubravka’s crossbar.
Josh Brownhill then found McNeil on the edge of the box, but Dubravka held onto the low strike.
Pope came to Burnley’s rescue in added time, dropping low to get a left-hand on Allen Saint-Maximin’s side-footed shot inside the box, after he had gone one on one with the Clarets stopper.
Second-half
Jackson introduced Wout Weghorst at the start of the second-half in place of Roberts.
Early on, Pope had to be alert to prevent a Newcastle chance, when Collins’ nearly turned Murphy’s cross beyond his own goalkeeper.
Up the other end, Cornet raced onto a pass, however Dan Burn’s well-timed challenge sent the ball out for a corner.
Newcastle’s second came on the hour. Wilson escaped Charlie Taylor to drill beyond Pope after Saint-Maximin’s driven cross into Pope’s bottom right-corner.
Burnley looked for a response, and captain James Tarkowski glanced a header wide of Dubravka’s upright.
The Clarets halved the deficit on 69 minutes after Cornet thumped a powerful left-footed shot from close-range.
Aaron Lennon came on for the Clarets with just under 15 minutes left, as Burnley continued to put the pressure on.
Buoyed by their goal, Burnley went in search of an elusive equaliser, with Barnes’ cross dropping to Weghorst who steered an effort inch wide.
Jack Cork then had a shot blocked prior to Dubravka thwarting added time efforts from Cornet and Barnes.
Despite their best efforts in the closing stages, Burnley were relegated from the top-flight.
Burnley: Pope, Roberts (Weghorst 45), Collins, Tarkowski ©, Long, Taylor, Cork, Bownhill (Lennon 73), McNeil, Cornet, Barnes
Subs not used: Hennessey, Bardsley, Thomas, Stephens, Dodgson, Costelloe, McGlynn
Newcastle: Dubravka, Trippier, Lascelles ©, Burn, Targett, Longstaff, Guimaraes, Joelinton (Murphy 9), Almiron (Ritchie 89), Wilson, Saint-Maximin (Wood 71)
Subs not used: Darlow, Dummett, Schar, Krafth, Fernandez, Gayle
Goals: Wilson penalty '20, '60, Cornet, '69
Referee: Craig Pawson