Almirón followed up his first goal of the campaign at Sheffield United with a spectacular opener before Isak scored his fifth of the season from the penalty spot in the second half as the Magpies maintained their momentum with another victory.
Injuries to Callum Wilson and Sven Botman meant that Isak and Jamaal Lascelles - as well as goalkeeper Nick Pope - stayed in the starting XI from the midweek Carabao Cup win over Manchester City, but there were eight changes to Eddie Howe's team for their return to league action.
Burnley, still looking for their first top-flight win of the campaign, came close to opening the scoring in the fourth minute when Luca Koleosho, who made a lively start to the game, took advantage of a Dan Burn slip to streak down the right. He cut the ball back for Zeki Amdouni but Pope made a fantastic save against his former club, blocking with his outstretched right leg.
After that let-off, the Magpies began to settle into the game and in the 11th minute, Bruno Guimarães punched the ball into the feet of Isak who turned and shot, but James Trafford was able to shovel it away.
But the visiting goalkeeper was unable to do anything three minutes later as Almirón opened the scoring.
Kieran Trippier dispossessed Aaron Ramsey deep inside the Clarets half and Almirón collected the ball and unleashed an arcing effort from 22 yards out which flew into the top corner of the Leazes net for his second goal in his last two league games.
In the 21st minute, Isak hooked the ball on for Sean Longstaff, who dragged his shot across goal, then seven minutes later Guimarães played a sumptuous through ball for Isak, who raced in on goal. Trafford attempted to smother at the Sweden striker's feet, but the ball bounced kindly for the Newcastle man - but, off balance, he was unable to turn the ball into the empty net.
United kept on pushing for a second and it almost arrived in the 35th minute as Longstaff, popping up on the right wing, looked up and picked out Anderson with a chipped cross. Anderson met the delivery with a diving header which looked certain to result in a first goal for his boyhood club, but Trafford saved with his feet and Connor Roberts hacked the ball away before it could spin over the line.
Burnley had an opportunity shortly before the break when Sander Berge broke and found Jordan Beyer, but the German's shot was deflected behind.
In first-half stoppage time, Anderson exchanged passes with Guimarães and let fly but Trafford got down to save, then with the first attack of the second period a seemingly goalbound drive from Trippier was diverted behind by Beyer.
Anthony Gordon sent a shot straight at his former England under-21 colleague and fellow UEFA European Under-21 Championship winner Trafford in the 73rd minute before, two minutes later, the United wideman was brought down in the box by Ameen Al-Dakhil.
Isak stepped up to send Trafford the wrong way and seal the three points for Newcastle, although they needed a block from Longstaff to deny substitute Wilson Odobert and a Pope save to thwart Al-Dakhil from the resulting corner to avoid a tense final few minutes.
A fifth booking of the season for Gordon - who will now serve a one-match ban at West Ham next weekend - and the sight of Joelinton limping off four minutes after coming on as a second-half substitute might take some of the gloss off the victory for Howe.
But his side recorded a fifth clean sheet in succession and a third straight league win as they prepare for their first Champions League home game in 20 years against Paris Saint-Germain on Wednesday night.
