Five-times World Cup winners Brazil lost 1-0 to Peru to exit the Copa America at the group stage for the first time in nearly 30 years.
Brazil had not been beaten by Peru in 16 previous meetings and needed just a point to reach the quarter-finals.
But substitute Raul Ruidiaz scored a 75th-minute winner at Foxborough Stadium in Massachusetts, with Brazil complaining it went in off his hand.
Ecuador beat Haiti 4-0 to make the last eight for the first time since 1997.
Brazil failed to score against Peru or Ecuador either side of an emphatic 7-1 victory over Haiti.
Asked if he feared for his job, Brazil boss Dunga said: “I am only afraid of death, I am not afraid of that.
“When you work for the Brazil national team you have to know the criticism will mount when you don’t get results, but internally we know what we are doing.”
Dunga lamented the awarding of the goal by Ruidiaz, who appeared to use his arm to bundle home Andy Polo’s cross.
Brazil’s players protested with Uruguayan referee Andres Cunha but after he consulted with the fourth official, he decided to stand by his decision to award the goal.
Dunga said: “It was such a clear handball, but we have no influence over the referee.”
The former World Cup-winning captain, who took over after Brazil’s 7-1 defeat by Germany in the 2014 World Cup semi-final in Belo Horizonte, still expects to be leading his country at their home Olympics in Rio de Janeiro in August.
Peru’s victory won Group B and will now play Colombia seeking to make a second Copa America semi-final in successive years, while Ecuador face the tournament hosts.