The Inter Milan midfielder headed in the crucial goal 22 minutes from time to extend Bert van Marwijk's stunning unbeaten record to 24 games after a bad-tempered encounter in which Brazil midfielder Felipe Melo was dismissed.
It was only the Netherlands' first win over Brazil since 1974 when the legendary Johan Cruyff inspired an Oranje victory.
But Sneijder now has the wonderful opportunity of adding the World Cup to his medal collection this season, after already winning the Champions League, Coppa Italia and Serie A.
Brazil's bitterly disappointing performance now means coach Dunga, who is not popular with the press back home, will face serious questions over his own future after failing to guide his country past the quarter-final stage.
Dunga cannot have envisaged the outcome early on. The Netherlands had not conceded a goal in open play before this game - but that proud record was shattered after just ten minutes.
Andre Ooijer was called up into the starting XI after Joris Mathijsen injured a knee in the warm-up, and his lack of communication with John Heitinga allowed Melo's wonderful pass to find Robinho who swept home the ball with ease.
Brazil had clearly done their work on danger man Arjen Robben, with the former Chelsea winger enduring rough treatment in a frustrating first half.
But the South Americans could have increased their lead when Juan somehow blazed over from six yards before Kaka tested Maarten Stekelenburg from outside the area.
The Netherlands were level though within seven minutes of the second half as Cesar committed the latest goalkeeping gaffe of the tournament, somehow failing to clear Inter Milan team-mate Sneijder's deep cross and watching it sail into the net via Melo's head.
The goal clearly unsettled Brazil and they fell behind 22 minutes from time - and this time there was no doubting that Sneijder would be claiming the goal. Robben's corner was flicked on by Dirk Kuyt and Sneijder was presented with the easiest of opportunities at the far post to find the corner with a header.
Brazil's chances were virtually dead and buried when Melo was given a straight red for a reckless stamp on Robben and there was to be no defiant fight back from the South Americans.
Netherlands (4-5-1): Stekelenburg, Van der Wiel, Heitinga, Ooijer, van Bronckhorst; Robben, Van Bommel, Sneijder, De Jong, Kuyt; Van Persie (Huntelaar 87).
Subs (not used): Vorm, Boschker, Boulahrouz, Braafheid, De Zeeuw, Schaars, Afellay, Van der Vaart, Elia, Babel.
Brazil (4-5-1): Cesar; Maicon, Lucio, Juan, Bastos (Gilberto 62); Alves, Silva, Kaka, Melo, Robinho; Fabiano (Nilmar 77).
Subs (not used): Gomes, Doni, Luisao, Thiago Silva, Gilberto, Josue, Julio Baptista, Kleberson, Grafite.