Goals from Jamaal Lascelles, Moussa Sissoko and Andros Townsend gave Rafa Benitez's side a hard-fought 3-0 victory at St James's Park this afternoon.
The result, Benitez's first as United manager, means the second-bottom club is now just three points adrift of safety.
Newcastle also have a game in hand over 17th-placed Norwich City, beaten 3-0 by third-bottom Sunderland earlier today.
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But the performance wasn't as convincing as the scoreline suggested.
And the crowd had a nervy 90 minutes.
Benitez, celebrating his 56th birthday, made five changes for the game – Jonjo Shelvey and Aleksandar Mitrovic were among those dropped – and he surprisingly handed the captain's armband to Sissoko.
Newcastle started positively, but struggled to play a telling ball in the final third of the pitch.
And mistakes crept into their game in a nervy first-half.
But Lascelles – who had called for more "heart and desire" on the pitch from United after last weekend's defeat at Southampton – lost his marker to head an Andros Townsend corner past Lukasz Fabianski five minutes before the break.
Swansea fought back after the break, and Newcastle had an uncomfortable spell on the back foot.
Newcastle's second goal, which came from another corner, settled the nerves inside the stadium.
Sissoko was the first to react after a flag kick was headed down, and he beat Fabianski with a close-range finish.
Mitrovic came off the bench to set up United's third goal, which came from a simple Townsend finish.
Newcastle could now climb out of the bottom three win over Manchester City at St James's Park on Tuesday night.
NEWCASTLE UNITED: Darlow, Anita, Mbemba, Lascelles, Dummett, Townsend, Tiote (Shelvey, 66), Colback, Sissoko, Wijnaldum (Perez, 73), Cisse (Mitrovic, 80). Subs not used: Woodman, Obertan, Aarons, de Jong.
SWANSEA CITY: Fabianski, Rangel, Fernandez, Williams, Taylor, Fer (Barrow, 57), Cork, Sigurdsson, Montero, Ayew (Gomis, 71), Paloschi (Routledge, 84). Subs not used: Nordfeldt, Naughton, Amat, Britton.
Goals: Lascelles 40 Sissoko 82, Townsend 90
Bookings: Fer 11, Mbemba 22, Anita 73
Referee: Lee Mason (Bolton)