Peter Bosz's men were trailing the Rotterdammers by six points heading into the encounter and knew defeat would effectively end their title ambitions.
The home side was clearly aware of the importance of the game and needed just 53 seconds to open the scoring via Schone. The Denmark international tried his luck from a free-kick well outside the area and beat goalkeeper Brad Jones with a stunning shot into the top corner.
Ajax remained well in charge after the early opener and doubled its lead in the 36th minute through Neres. The Brazilian youngster - who made the starting XI for the first time since his big-money move from Sao Paulo - beat his marker to Hakim Ziyech's cross from the left and coolly slotted home from close range.
Bertrand Traore should perhaps have made it three with 10 minutes left after being sent clean through by Ziyech, but Jones proved to be too much of an obstacle.
Michiel Kramer pulled one back well into stoppage time with a powerful shot from outside the box, but it was too little, too late as Ajax held on.
This weekend's results mean Ajax have pulled to within three points of Feyenoord with six games left, with PSV trailing the leaders by five points in third place.