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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9. Ryan Fraser - 5
Got into some dangerous areas but often lacked the quality with the final ball. Photo: Shaun Botterill
10. Joelinton - 5
Clipped an early shot wide and won the ball back well on a couple of occasions before going off injured in the first half. Photo: Ian MacNicol
11. Allan Saint-Maximin - 6
Dragged a shot wide in the first half. Typically lively but ultimately ineffective. Booked. Photo: Alex Livesey
12. Chris Wood - 5
Won a couple of flick ons but left isolated once again for large periods. The times the ball did get to him, his touch let him down. Photo: PAUL ELLIS