A goal from Stephane Sessegnon was enough for the Black Cats to claim all three points at the Stadium of Light as they boosted their chances of survival.
Meanwhile, the 1-0 defeat has all but ended Everton's hopes of qualifying for next season's Champions League.
"Disapointing result we never got going today and Sunderland fought hard. Still 4 games left and we will fight to the end," Neville wrote on Twitter.
Next weekend Neville's team will play host to Fulham.