Brazil beat Chile 1-0 after losing Gabriel Jesus to a redcard. Paraguay was beaten on penalties by Peru, who drew a 3-3 draw.
In the semi-finals, Peru will face the host nation at Rio de Janeiro’s Nilton Santos stadium.
On Saturday, the other quarter-finals are played with Argentina facing Ecuador and Uruguay facing Colombia.
Friday's tie between Copa America champions was closer than their form suggested.
Brazil won 11 of the 11 games, topping Group B. Chile was fourth in Group A. They have drawn four of their six last matches 1-1.
After an uneven and goalless 45 minutes, the match ignited within seconds of the restart. Substitute Lucas Paqueta worked with Neymar in a scrappy one to two and then held off Gary Medel who fired home from close range.
The outlook was changed in less than two minutes. Brazil was reduced to 10 men when Gabriel Jesus was given a straight red for his horrific kung fu kick that sent Eugenio Mea tumbling.
Image: Gabriel Jesus was sent to Brazil for a kung fu kick. However, they beat Chile 1-0 in quarter-finals
Brazil is used to being under pressure at home and Brazil was not prepared for Chile. Chile forced Brazil onto the back foot during the second half, when they tried to find an equaliser.
Alexis Sanchez was replaced at half-time by Ben Breton, an English-born striker. Ederson made at most two good saves.
The Chileans, who were the winners in 2015 and 2016, couldn't achieve the goal and lost.
In the most exciting tournament match, Paraguay beat Peru 4-3 with penalties.
After a difficult match, the game ended 3-3.
After Peruvian goalkeeper Pedro Gallese had denied Alberto Espinola during the shootout, it was the decisive moment. Miguel Trauco converted his kick to bring the Copa America 2019 runners-up back into contention.
Braian Samudio and Daniel Martinez were also absent from Paraguay. Antony Silva of Paraguay kept Santiago Ormeo, Christian Cueva out.
Image: Peru's team celebrates their Copa America quarterfinal victory over Paraguay
Paraguay scored in the 11th minute, Gustavo Gomez scoring from a corner.
Lapadula scored his first goal of the game from close range in the 21st minutes, assisted by Andre Carrillo (ex-Loanee Watford).
After Paraguayan defenses stopped complaining about a foul, he added another in the 40th minutes with a cross-cum shot from the left flank.
After Peru took the lead, Peru seemed certain that they would progress. Paraguayan defender Gomez received an additional yellow card and was sent off.
Paraguay began the second half with a strong start and managed to exert pressure on Peru. Junior Alonso scored the equalizer in 54 minutes after he fired the ball past Gallese at another corner.
Peru's Yoshimar Yotun scored a goal at the box's edge 10 minutes after the ball was deflected off of a Paraguayan player. But there were more twists to come.
Five minutes later, Andre Carrillo, a Peruvian striker, also received his second yellow. The two sides were now 10-vs-10.
Paraguay now throws everything at Peru. Substitute Gabriel Avalos scored a last-minute goal to tie the match and send it off to penalties.