The 30-year-old returned to the starting lineup after missing two games with an ankle injury, with head coach Tite reverting to his strongest side after making wholesale changes from the 1-0 group-stage defeat to Cameroon last Friday.
Vinicius Junior opened the scoring in just the seventh minute with a composed curing finish, before Neymar netted his first goal of the tournament in Qatar from the penalty spot six minutes later, after Richarlison was clipped inside the box.
Indeed, Richarlison was involved in Brazil's third goal of the night just before the half-hour mark, finishing a brilliant free-flowing team goal from close range, before Lucas Paqueta volleyed in their fourth in the 36th minute.
A sweetly struck volley from substitute Paik Seung-ho was nothing but a consolation for South Korea in the final 15 minutes, and Brazil eventually saw out the game to book their place in the quarter-finals.
Neymar has now notched up 76 goals in 123 international appearances for Brazil, moving just one behind the nation's all-time leading scorer Pele, who scored 77 times for the Selecao.
The Paris Saint-Germain attacker has also become just the third Brazilian player alongside icons Pele and Ronaldo to score at three different World Cup tournaments.
Neymar has scored seven times in 12 World Cup appearances, including four in 2014, two in 2018 and his strike against South Korea in this year's tournament.
In comparison, Pele scored at four editions of the World Cup, netting 12 times in 14 games between 1958 and 1970, while Ronaldo scored 15 goals in 19 matches between 1998 and 2006.
Neymar has revealed that he was "crying a lot" after sustaining an ankle injury in Brazil's opening group game against Serbia, and has admitted that he was "scared" at the prospect of missing the rest of the World Cup.
The PSG attacker struggled with injuries at the previous two World Cup, having previously missed the 2014 semi-finals with a back problem, while he was not fully fit to represent Brazil at the 2018 tournament due to a foot injury.
However, Neymar is pleased to have recovered in time for the knockout rounds and believes Brazil are in a strong position to go all the way and win a record-extending sixth World Cup this year.
"I was very scared because I was doing very well, after a very good season (with PSG). Suffering an injury like I did is very hard," Neymar told Globo after the game. "I spent the night of the other day crying a lot. My family knows, but it all worked out. It was worth the effort.
"Surely the time has come [for Brazil] to go further. We dream of the title, but we have to go step by step. It was the fourth game today, three to go. We are very prepared, with our heads focused on getting this title."
Neymar was not the only Selecao star making history at Stadium 974, as centre-back Thiago Silva has now drawn level with Cafu and Dunga in captaining the national team on 11 occasions at the World Cup.
At the age of 38 years and 74 days, the Chelsea defender has also become the oldest player to provide an assist on the biggest international stage since Cameroon legend Roger Milla (42 years and 39 days) after setting up Richarlison for Brazil's third goal.
The South American giants will now prepare for Friday's quarter-finals clash against Croatia, who beat Japan 3-1 on penalties to progress from the last 16.