The win moves Ajax onto seven points with one game remaining and the club's hopes of progressing to the last 16 rest on beating AC Milan at San Siro, while defeat for the Blaugrana was the club's first of the season and denied coach Gerardo Martino a share of Pep Guardiola's club record of 21 games unbeaten.
The visitors made four changes, with Carles Puyol, Martin Montoya, Alex Song and Xavi all starting for Martino's men. The South Africa international ghosted into the penalty area to meet Ricardo van Rhijn's cross from the right and the 23-year-old calmly side-footed the ball past Jose Manuel Pinto.
After Lasse Schone had missed two chances, Ajax finally doubled their lead just before the break through Hoesen. Pinto saved well from Viktor Fischer and Hoesen latched onto the loose ball, lifted it past Puyol and fired into the bottom corner. The game was turned on its head four minutes after the break as Van Rhijn played a loose pass back to Veltman. Neymar pounced but he was brought down by the defender just inside the area.
Jasper Cillessen kept the hosts in the game with a good save from Neymar on the hour and a resilient Ajax defence managed to hold on for a vital win. Despite a confident start, Ajax were lucky to survive a swift Barcelona counterattack in the 14th minute, as Veltman made a crucial block to deny Cesc Fabregas.
