Townsend added the third with one minute remaining, sliding in Aleksandar Mitrovic's pass to give Rafa Benitez his first win since becoming Newcastle's boss five weeks ago.
Although they remain in the relegation zone, they are only three points behind 17th-placed Norwich with a game in hand.
Swansea dominated much of the play after the break but Ashley Williams and Jefferson Montero missed good scoring chances.
(Reporting by Mike Collett; Editing by Toby Davis)
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