It was the second tournament in succession that Paraguay have beaten Brazil on penalties at the quarter-final stage after 2011 when they went on the reach the final.
The face Argentina on Tuesday for the second time in the tournament, having fought back from two goals down for a 2-2 draw in the teams' opening group match.
Gonzalez netted the decisive Paraguayan penalty with the fifth spot kick having scored his team's equaliser during normal time.
Robinho started and finished a fine move in the 15th minute to put Brazil ahead before Gonzalez equalised with 20 minutes to go when he converted a penalty after a hand ball by Thiago Silva.
“Unfortunately, our game dropped in the second half,” said Robinho. “We had chances to kill the game off.
“Paraguay with all respect aren't one of the best sides. We didn't kill the game and unfortunately we lost a goal with a silly mistake and we lost.”
Brazil, who had won 12 of their previous 13 matches since Dunga took charge after last year's World Cup, had more possession in the first half and weaved some good attacks but Paraguay were quick and dangerous on the break.
Paraguay became more adventurous in the second half and winger Edgar Benitez soon won a free kick when he was brought down by Dani Alves, though goalkeeper Jefferson parried the ball away from Roque Santa Cruz's attempt.
With an hour gone, Jefferson made a diving reflex save from centre back Paulo Da Silva's header and later made a similar save from Gonzalez in a counter-attack.
Paraguay captain Justo Villar then had to make a scrambling save from a Philippe Coutinho shot as Brazil piled on late pressure.
In the shootout, the writing was on the wall for Brazil from their second penalty which was put wide by Everton Ribeiro and another substitute, Douglas, missed their fourth to give Paraguay two penalties to settle the outcome.
Santa Cruz then skied his penalty, Philippe Coutinho gave Brazil hope by netting his but Gonzalez made no mistake.
Hosts Chile meet Peru in Monday's other semi-final. – Reuters
Original source: Paraguay down Brazil on penalties