Callum Wilson’s seventh goal of the season from the penalty spot gave Newcastle a 1-0 lead going in at half-time.
Wilson then doubled The Magpies’ lead in the second half with a simple tap-in from Allan Saint-Maximin’s smart pass.
Maxwell Cornet then pulled a goal back for Burnley as they fought to remain in the Premier League but it wasn’t enough as the game ended 2-1.
The result confirmed Burnley’s relegation to the Championship as Leeds United survived the drop.
Eddie Howe made two changes to the side that beat Arsenal 2-0 at St James’s Park on Monday night with Kieran Trippier and Jamaal Lascelles coming in for Emil Krafth and Fabian Schar.
It was Trippier’s first start for Newcastle in over three months as he faced his former club following after recovering from a broken metatarsal.
9. Miguel Almiron - 6
Angered Wilson after going for goal from a tight angle. Kept going but a quiet afternoon on the whole. Photo: Stu Forster
10. Allan Saint-Maximin - 7
Could have made it 2-0 on the stroke of half-time but did well to set up Wilson’s second of the afternoon. Photo: Ian MacNicol
11. Callum Wilson - 9
Did Leeds a big favour as he calmly converted his seventh goal of the season from the penalty spot. Added another from close range in the second half.
12. Jacob Murphy - 6
On for Joelinton 11: Put in a decent cross which was almost turned in by Collins. Was kept busy down the right side. | AFP via Getty Images