Emile Smith-Rowe put Arsenal ahead at Eppleton Colliery Welfare Ground with a curling strike before 16-year-old Tyreece John-Jules doubled the lead.
After the break Sunderland pulled a goal back through Elliott Embleton, but within two minutes a second goal from John-Jules sealed a fourth win in five matches.
The defeat leaves Sunderland two points above the relegation zone.
Sunderland: 5 Feb v West Ham (A) PL2
Arsenal: 5 Feb v Everton (H) PL2