The 31-year-old went into the World Cup qualifier level with Pele on 77 goals and knowing that he only needed one more to break a record which had been held by the legendary striker since 1971.
Neymar squandered a golden chance to reach that historic mark when he missed a penalty after 17 minutes, but the Al-Hilal man made amends with his 78th international strike shortly after the hour mark.
Neymar then added his second of the night - and Brazil's fifth overall - in stoppage time to take his tally up to 79 goals in 125 appearances.
"I never imagined reaching this record. I'm no better than Pele or any other player for the national team," he said after the match.
Brazil coach Fernando Diniz added: "He came to do what he did: have fun, score two goals and break the record.
"He is a very big idol, people must recognise him and accept him."
Pele took 92 games to score his 77 goals for Brazil and is regarded as one of the greatest footballers of all time, having won the World Cup a record three times.
Neymar is still some way short of becoming the all-time top-scorer in Brazil's football history, with Marta having netted 122 goals in 189 games for the women's team.
Brazil's win takes them to the top of the CONMEBOL World Cup qualifying standings after one game each, joining Uruguay, Argentina and Colombia in taking maximum points from their opening match.
Also on Friday night, Marcelo Bielsa won his first game as Uruguay coach courtesy of a 3-1 triumph over Chile, with Real Madrid's Federico Valverde among the goalscorers.