After a goalless first half which saw Joelinton go off injured and Jonjo Shelvey force a smart save from Leeds goalkeeper Illan Meslier, the Magpies number 8 found the breakthrough with 15 minutes of normal time remaining from a free-kick.
The win leaves Newcastle 18th, just a point from safety with a game in hand on 17th placed Norwich City.
Their next Premier League match will be against Everton on February 9. The managerless Toffees sit just four points ahead of Newcastle in the table.
Eddie Howe made just one change from the previous match as Joe Willock came into the side for Sean Longstaff while Javier Manquillo returned from suspension with a place on the bench.
Here are Dominic Scurr’s player ratings from the match…
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13. Sean Longstaff - 7
On for Joelinton 42: Shot just wide from the edge of the box. A solid outing from the bench. Photo: Ian MacNicol
14. Javier Manquillo - 7
On for Dummett 63: Lively off the bench and won the free-kick that resulted in Newcastle's opener. Photo: PAUL ELLIS
15. Ciaran Clark - 6
On for Lascelles 69: Helped Newcastle see out their first away clean sheet of the season with minimal fuss. Photo: Stu Forster