Puig, ranked 34 in the world, said the victory against third seed Muguruza was the biggest win in her career.
"I still have goosebumps," said Puig, who will play Germany's Laura Siegemund in the quarterfinals.
"I already know where I want to get and I'm letting the world know that I've arrived."
Muguruza, who lost in the second round at Wimbledon, said she was surprised by the intensity and quality of play by 22-year-old Puig.
"I was very shocked about how she played today. I think she played unbelievable. Also I didn't find the way tactically to beat her," she said.
In another upset, Russia's Evgeny Donskoy edged out Spain's world number 12 David Ferrer in three tense sets. Donskoy, ranked 84 in the world, put out former French Open finalist Ferrer 3-6 7-6(1) 7-5.
"In important moments he played more aggressively than me," said 34-year-old Ferrer, who conceded this may be his last Olympics.
The highest seeds remaining in the men's tournament, Andy Murray and Rafael Nadal, progressed into the third round with ease.
Murray, who won singles gold at his home Games in London, outclassed Monaco 6-3 6-1 and will next face either France's Benoit Paire or Italy's Fabio Fognini.
"I played pretty well, hit the ball good from the back of the court, not many unforced errors. It was a good match," said Murray, who is favourite defend his title after Serbian world number one Novak Djokovic was eliminated in the first round.
The 2008 Olympic gold medallist Nadal overpowered Italy's Seppi Andreas 6-3 6-3 despite admitting over the weekend that he was still not fully fit.
"It was important to be playing like this. I'm enjoying [myself], I'm happy on the court," the Spanish third seed said.
His compatriot Carla Suarez Navarro was less fortunate. World number nine Madison Keys outlasted her 6-3 3-6 6-3 to reach the last eight.
Keys struggled to find her rhythm but in the end her powerful strokes proved too much for Suarez Navarro, who has one of the best one-handed backhands in women's tennis.
"It was a really tough match," said Keys, who will now face Russia's Daria Kasatkina.
