Ajax held on under plenty of pressure in the latter stages to frustrate the visitors.
Below, Sports Mole takes a look at the key talking points on a dramatic evening in Amsterdam.
On target 5
On target 2
Was the result fair?
It's difficult to see how the biggest Barcelona fan can argue with the result this evening. Ajax were completely dominant in the first half and when they were asked to show a different side to their play in the second half, they certainly passed the test. Barcelona applied plenty of pressure in the closing stages but they struggled to create clear chances.
Frank de Boer will be absolutely delighted with his players this evening. In the first half we saw the type of football that Ajax were famous for playing back in their glory days and the young team showed a great attitude when they were reduced to 10 men. We saw it all from the hosts tonight. When they attacked, they did so in great numbers but they also displayed impressive work ethic in defence.
Barcelona simply didn't produce anything near their best. It could be argued that the visitors played with a relaxed mentality after already securing their spot in the next round but Gerardo Martino still won't be happy. They struggled to keep the ball in midfield in the first half and chances were rare even when Veltman was sent off.
Sports Mole's man of the match:
Daley Blind: The 23-year-old stood out for the hosts tonight, but there were a number of players who could have won this award. Blind started the game in central midfield and he impressed by getting the better of Xavi and Andres Iniesta. However, it could be argued that he was even better when moved to left-back as he kept Pedro quiet with a faultless display.
Ricardo van Rhijn made things very difficult for his team when his lazy pass gifted the ball to Neymar, who was then brought down by Veltman. Van Rhijn's teammate was dismissed but he'll be the first person to offer an apology. Luckily for Ajax, it didn't cost them a famous win.
Pavel Kralovec did a good job of keeping a quick game flowing despite being forced to dish out a number of cards for cynical challenges. However, Ajax were left frustrated by his decision to point to the spot when Veltman appeared to tackle Neymar on the edge of the box.
What next? Ajax: De Boer's side will be full of confidence when they face ADO Den Haag in the league on Sunday. Barcelona: The Catalan club will look to get back to winning ways when they travel to the Basque region to play Athletic Bilbao on Sunday night.
Ajax: De Boer's side will be full of confidence when they face ADO Den Haag in the league on Sunday.
Barcelona: The Catalan club will look to get back to winning ways when they travel to the Basque region to play Athletic Bilbao on Sunday night.