The 31-year-old striker went into the game level with Pele on 77 goals and had a golden opportunity to break the record in the 17th minute when Brazil were awarded a penalty.
But he looked distraught after his soft spot-kick was easily saved by Bolivia keeper Guillermo Viscarra to keep the scores goalless.
Seven minutes later, Brazil got in front when Rodrygo reacted quickly to slot home after a shot from Raphinha was parried out by Viscarra.
Brazil doubled their lead just after the break when Raphinha cut in from the left and his low shot took a slight deflection as it flew into the bottom corner.
Neymar had a hand in Brazil’s third, with a clever chipped pass that was flicked on by Bruno Guimaraes into the path of Rodrygo who confidently drilled home.
Then came Neymar’s moment to enter the history books as he pounced on a loose ball inside the box and tucked away his 78th goal in his 125th game for his country – raising his hands and his gaze to the skies.
Victor Abrego pulled a goal back for Bolivia, bustling through a disorganized Brazil backline before blasting an unstoppable shot past Ederson.
But there was still time for Neymar to extend his record tally, connecting with a low ball from Raphinha in stoppage time to complete the victory and a memorable night for the forward.
The 31-year-old former Barcelona striker recently moved from Paris Saint-Germain to Saudi Arabian club Al Hilal.
Pele, who died in December aged 82, scored 77 goals for Brazil in 92 appearances between 1957 and 1971.
The win was the perfect start for new coach Fernando Diniz, who took over from predecessor Tite in the wake of the Selecao’s World Cup quarter-final exit last year.
Diniz paid tribute to Neymar, whose role in the squad has been under scrutiny.
“Many did not know what he would come to do, and he came to do what he did: have fun, score two goals and break the record,” he said.
“He is a very big idol, people must recognize him and accept him,” he added.
In Friday’s other game, Uruguay turned on the style, under new coach Marcelo Bielsa, as they beat Chile 3-1.
Argentine Bielsa, who had previously coached Chile between 2007 and 2011, opted for a youthful line-up without veteran strikers Edinson Cavani and Luis Suarez.
Bielsa was rewarded with an entertaining display from his new charges with River Plate attacking midfielder Nicolas de la Cruz scoring twice.
De la Cruz opened the scoring with a fine left-foot strike in the 38th minute before Real Madrid midfielder Federico Valverde doubled the advantage on the stroke of half-time.
De la Cruz broke down the left before switching to Valverde, who made space before burying a shot into the far bottom corner.
The third goal, in the 71st minute, came after a fast break down the left with Liverpool striker Darwin Nunez pulling the ball back into the path of De la Cruz who made no mistake from close range.
Chile pulled a goal back three minutes later with Arturo Vidal scrambling the ball home after a Diego Valdes header came out off the bar.
Bielsa was delighted with the display from his new charges.
“I like the style of the Uruguayan players,” he said, “The style of play is imposed by individuals not the coach. The first ten minutes were not good but then we progressively generated goal opportunities and wore down the opponent,” he added.
The qualification process for the expanded 48-team World Cup, to be held in the United States, Canada and Mexico in 2026, offers two extra places for South America with six teams qualifying directly.
The seventh out of the 10 nations will head to an inter-continental play-off.
The opening games on Thursday saw Argentina beat Ecuador 1-0 with a goal from Lionel Messi while Peru drew 0-0 at Paraguay and Colombia beat Venezuela 1-0.
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