The 20-year-old netted from the penalty spot, while laying on goals for Elano and Paulo Henrique Ganso before Alan Kardec struck before the interval. Elano and Ganso added to their personal tallies after the break, with Borges sealing the rout.
The result earned Santos a 9-2 aggregate victory after their 2-1 first-leg loss in La Paz, and set up a last-eight meeting with Argentinian side Velez Sarsfield.
Neymar's goal took him to 105 goals for the club, making him the team's top scorer in the post-Pele era.
The delighted youngster said: "I'm living the best year of my career. We showed that we are a very powerful team playing at Vilha Belmiro. That's because our people are happy to once again experience big moments in the club's history."
Muricy Ramalho's team will travel to Buenos Aires for the first leg against Velez next week.
Meanwhile, in Santiago, Copa Sudamericana champions Universidad de Chile thrashed Deportivo Quito 6-0 to overturn a 4-1 first-leg deficit.
Junior Fernandes and Angelo Henriquez notched two goals each, with Eugenio Mena and Marcelo Diaz completing the scoring for Jorge Sampaoli's team. The win set up a meeting with Paraguayan club Libertad, who eliminated Mexican side Cruz Azul.
Finally, in the all-Brazilian last-16 duel, Fluminense beat Internacional 2-1 in Rio de Janeiro to progress by the same scoreline on aggregate and secure a date with Argentinian giants Boca Juniors.
Leandro Damiao had put the Porto Alegre-based team ahead in the 14th minute, but Abel Braga's Fluminense turned things around before half-time thanks to goals from Leandro Euzebio and Fred.