A controversial goal sealed it for the South American side in the first half. A dubious free-kick was awarded by a linesman, from which the Uruguayan's grabbed the only goal of the game.
Schwarzer played for the full 90 minutes but Viduka had to settle for a role as a second half substitute.
Szilard Nemeth's Slovakia have a mountain to climb after a 5-1 defeat in Madrid. Spain led 2-0 at half time but Nemeth reduced the arrears early in the second half.
The turning point of the game came in the 64th minute when Slovakia were reduced to ten men. Spain scored from the penalty spot and all but guarenteed their ticket to Germany by running in two further goals.
George Boateng earned his third cap for Holland, after coming on for the last four minutes, in Holland's 3-1 defeat by Italy in Amsterdam.