Paraguay repeated their 2011 Copa America penalty shootout win over Brazil to set up a semi-final meeting with Argentina.
Former Manchester City striker Robinho scored on his 99th appearance to give Brazil a 15th-minute lead as he tucked away a Dani Alves cross.
But Thiago Silva’s second-half handball allowed Derlis Gonzalez to equalise from the spot and force the shootout.
Brazil then missed two attempts and went out as Gonzalez again converted.
The defeat curtailed Brazil’s revival under Dunga following their 7-1 semi-final defeat by Germany in last summer’s World Cup finals in their own country.
The eight-times winners have not won the Copa America since 2007, while Paraguay now have the chance to go on and lift the trophy for the first time since 1979.
They play Argentina in Concepcion at 00:30 BST on 1 July with the other semi-final seeing host country Chile meet Peru ahead of the final in Santiago on 4 July.
Brazil were unable to build on the perfect start given to them by Robinho – playing in place of the suspended Neymar – as the 31-year-old scored his first competitive goal for his country since 2010.
Robinho was helped by clever movement from Roberto Firmino, although that was virtually the only major contribution Liverpool’s new £29m signing from Hoffenheim made before being replaced in the second half.
Brazil barely created another chance, while Paraguay gradually stepped up the pressure in the second half.
Defender Paulo da Silva was denied by Jefferson’s save as he headed goalwards following a corner.
But an error of judgement from Paris St-Germain defender Silva gifted a deserved equaliser and Paraguay then held their nerve to progress with the tie going straight to penalties after 90 minutes.
Brazil did better than in 2011 when they failed to score a single penalty in the shootout but Paraguay could even afford a miss from former Manchester City and Blackburn striker Roque Santa Cruz as they recorded a memorable win.