The Netherlands will meet either Ghana or Uruguay, which played a quarterfinal match later Friday, in the semis Tuesday.
The Dutch, trailing at intermission, played with more urgency in the second half than Brazil and it paid off about halfway through the period.
With the score tied 1-1, Arjen Robben sent a Dutch corner kick into the middle of the box where Dirk Kuyt slightly changed the flight with a header. Sneijder was then able to solidly nod the ball into the left side of the net in the 68th minute.
About 5 minutes later, Brazil's Felipe Melo was sent off for stepping on an opposing player, leaving the Brazilians playing a man short the rest of the way.
Brazil broke through first with Robinho taking a long lead pass from Melo and breaking past the Dutch defense. Robinho then easily beat Netherlands goalkeeper Maarten Stekelenburg in the 10th minute.
The score was tied in the 53rd minute on what was ruled an own goal. Sneijder took a short free kick and tried a shot from long range. His high arching attempt just glanced off Melo, who may have also gotten in the way of Brazilian goalkeeper Julio Cesar, and the ball went into the goal.
It was the first own goal charged to Brazil in its World Cup history -- a record 97 games -- but the second time in the current tournament the Netherlands benefited from an own goal. One of the goals in the Netherlands' 2-0 Group E win over Denmark was an own goal.
In a signing ceremony at the White House, the president said one of his administration's highest priorities is to prevent the spread of nuclear weapons, and the United States and its allies have "strengthened the global non-proliferation regime, including the cornerstone of our efforts -- the Nuclear Non-Proliferation Treaty." He said Iran is the only signatory to the treaty "that has been unable to convince the International Atomic Energy Agency that its nuclear program is for peaceful purposes."
Following the session, Obama told reporters the pair discussed "issues related to Afghanistan and Pakistan; Iran and its attempts to develop nuclear weapons capacity."
"We discussed the Middle East peace process and the importance of moving forward in a significant and bold way in securing a Palestinian homeland that can live side by side with a secure and prosperous Israeli state," the president said.
During their talks, both leaders expressed strong support for getting Iran to "meet its international obligations" regarding its nuclear program and hope the proximity talks will lead to a resolution of the Middle East conflict.
Obama thanked Abdullah for his hospitality when Obama visited Riyadh and pointed out the historic ties between the United States and Saudi Arabia dating back to the meeting 70 years ago between Franklin Delano Roosevelt and King Aziz.
Abdullah echoed the sentiments and said he has heard from people around the world who consider the American people "friends of Saudi Arabia and its people and … friends of the Arab and Muslim people, and … also friends of humanity."
"One hundred members of the Senate sit here in the shoes of more than 300 million Americans as we discharge our duties," committee Chairman Patrick Leahy said. "The Supreme Court stands for all Americans ... it is an awesome responsibility."
If confirmed, President Barack Obama's second Supreme Court nominee -- now the solicitor general -- would be the fourth female justice and third woman sitting on the court, joining Ruth Bader Ginsburg and Sonia Sotomayor.