The Clarets are on 24 points from their 36 matches, and with Nottingham Forest beating Sheffield United 3-1 on the same afternoon, Vincent Kompany's side are now five points behind the 17th-placed Reds in the table.
Burnley need to win both of their final league games against Tottenham Hotspur and Forest, and then hope for favours from elsewhere, if they are to stand a chance of staying in England's top flight.
As for Newcastle, the victory has moved Eddie Howe's side above Manchester United into sixth position, and it was a brilliant performance from the visitors at Turf Moor.
Burnley started the brighter on home soil, with Jacob Bruun Larsen striking wide in the early exchanges, before the hosts had a shout for a penalty when Lorenz Assignon went down under contact from Bruno Guimaraes.
Burnley now face huge task to remain in Premier League
Burnley boss Kompany was convinced that it was a penalty, but referee Anthony Taylor said no and VAR agreed, much to the frustration of the Clarets.
There were cheers inside Turf Moor when news of Sheffield United taking the lead against Forest filtered through, but Burnley found themselves behind in the 19th minute of the contest, with Callum Wilson tapping into the back of the net from close range after Alexander Isak had been denied by Arijanet Muric.
Newcastle should have doubled their lead soon after when Wilson found Tino Livramento inside the penalty box, but the defender fired straight at Muric from six yards out.
Burnley just could not resist the pressure, and the Magpies made it 2-0 in the 36th minute through Sean Longstaff, who carefully picked out the bottom corner after being set up by Jacob Murphy inside the penalty box.
Two then became three in the 40th minute, with Guimaraes getting in on the act; the Brazilian sliding one into the back of the net after being picked out by Anthony Gordon.
Alexander Isak moves onto 20 Premier League goals for season
Burnley entered the match having lost just one of their last eight league matches, and they claimed a point in a 1-1 draw away to Manchester United last weekend.
The Clarets simply had no answer to Newcastle's quality on Saturday, though, and Kompany's side were perhaps a touch fortunate to only concede four on a very tough afternoon.
Lyle Foster did have a chance to register early in the second period, but he smashed over from close range, and Newcastle were then awarded a penalty when Josh Brownhill pulled Gordon's shirt inside the box.
Isak stepped up to the penalty spot, but the Sweden international was denied by Muric.
More chances were always likely to come, though, and Isak did get his name on the scoresheet just three minutes after missing his spot kick, as he turned a cross from Murphy into the back of the net to make it 20 Premier League goals for the season.
Burnley did give their fans something to cheer late on, with Dara O'Shea heading into the back of the net, but it was a case of too little, too late for the Clarets, not just this afternoon but also in their season.