The Magpies' first win under Rafael Benitez, at the fifth time of asking, came via goals from a couple of unlikely sources in either half as Jamaal Lascelles and Moussa Sissoko found the net, before Andros Townsend added a late third.
Just a seventh win of the season lifts United within touching distance of Norwich City, who were defeated by Sunderland earlier today, and ensures that the Toon Army's survival hopes remain alive and well.
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Lukasz Fabianski was called into action 10 minutes in to claw away a dipping cross from the right that was destined for the far corner of the net if not for his timely intervention.
The ball was only half cleared, though, and when it was worked to Townsend 25 yards out the England international only had one thought on his mind - taking on the shot, which Fabianski was equal to.
Newcastle were looking to capitalise on their opponents' sluggish start to the match, and they were so nearly rewarded for some smart play when Papiss Cisse flicked on Lascelles's ball into the path of Jack Colback.
Colback found himself through on goal, albeit at a tight angle which would ultimately work against him as he blasted into the frame of the Swans' keeper.
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Fabianski was again called into action six minutes before the break when keeping out Townsend's long-range drive for a second time, yet the breakthrough goal did arrive prior to the interval.
Lascelles proved to be the hero, simply brushing off marker Gylfi Sigurdsson to header home past Fabianski who will perhaps be disappointed not to have kept out the powerful attempt from close range.
Despite the hosts starting the second half the brighter of the two, it was Swansea who created the first real chance of a scrappy contest when Andre Ayew was picked out in space only to see his shot blocked by Chancel Mbemba.
City rallied and a second cross soon came into the opposition box, which a quiet Alberto Paloschi glanced narrowly wide of the far post on the hour mark.
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A rather audacious back-heel flick, which was not quite on target, was all the Magpies truly had to show for their attacking work in the second period as the match reached its final quarter.
Having looked so comfortable for large parts, United were suddenly left holding on when Swansea - who had won three on the bounce here prior to today's setback - created three big chances in quick succession.
The third chance fell the way of substitute Bafetimbi Gomis, whose first touch set him up perfectly for a shot which he somehow scooped over the bar from a promising position.
It was simply not to be for the visitors, however, as Sissoko turned home from close range late on when a couple of ricochets saw the ball fall nicely to him after a well-taken corner.
Townsend added the icing on the cake in the final minute of normal time, making the most of Aleksandar Mitrovic's pass to leave him one on one with Fabianski by picking his spot to seal a memorable win for Newcastle.