Venezuela, which has never won the Copa America or qualified for a World Cup, finished Group A second to Brazil.
Peru finished third after being hammered 5-0 by the Copa America host and will have to wait for other results to know if it advances as one of the best third-placed teams.
Venezuela's opponent in the quarterfinals will be the second-placed team in Group B, either Paraguay, Qatar or Argentina.
After two goalless draws with Peru and Brazil, Venezuela knew it had to bring its most attacking formation against Bolivia at Mineirao Stadium to advance. The team also had to work without its two central defenders, Yordan Osorio and Mikel Villanueva, one affected by injury and the other by a fever. Jhon Chancellor and Luis Mago stepped up.
Venezuela coach Rafael Dudamel said the team showed solidity by imposing its game against Bolivia.
"We had to work more in our defense than expected; it was a team that was hurt. And they did very well," Dudamel said. "We controlled (the match) tactically and we responded to match situations; that gives us the tranquility of knowing that we have a team and that everyone knows they are important."
Venezuela scored after only two minutes, when Machis headed a cross from Ronald Hernandez from the right. Only 4,640 people were at Mineirao Stadium to watch that moment.
The Vinotinto added a second in the 55th, Machis scoring from the edge of the box.
Leonel Justiniano reduced Bolivia's deficit in the 82nd, but Josef Martinez, who substituted Machis shortly before, secured the Venezuelan victory four minutes later with a header after a cross by Yeferson Soteldo.
Bolivia coach Eduardo Villegas was philosophical about his team's performance in its third loss at Copa America.
"We learned a lot. We came with a group of players of intermediary age and young men that earned some experience, saw what it is like to be in an international competition," he said.
Dudamel said the quarterfinals clash, no matter against whom, will inspire an even better Venezuela.
"We have shown ourselves as a more complete team each day, in every aspect," he said. "We have to keep this sequence; we will not do anything differently."
Venezuela's next match will be on Friday at Maracana Stadium in Rio de Janeiro.