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The star forward opened the scoring early in the second half at the Samara Arena and it was his shot being saved which led to Roberto Firmino's 88th-minute clincher.
Brazil will meet Belgium or Japan, in a quarter-final in Kazan on Friday.
It's the seventh consecutive quarter-final appearance for the five-time champions, while Mexico exit in the last 16 for the seventh straight World Cup.
“I would like to say once again that being able to play at such a level against a team like Brazil really shows Mexico is a good team,” coach Juan Carlos Osorio told reporters.
“We were not efficient and we didn't have that extra quality that they had.”
Both sides had early chances as Brazil keeper Alisson punched a cross straight to Hirving Lozano, whose shot was blocked by Miranda.
For Brazil, Neymar had an effort palmed away by the outstanding Guillermo Ochoa.
Lozano then just missed connecting with Javier Hernandez at the far post and had another shot blocked after a fine cross-field ball from Carlos Vela.
El Tri's confidence continued to grow as they worked down the left with Vela laying off for Hector Herrera, whose shot was snuffed out by Thiago Silva in the area.
Neymar then began Brazil's awakening by brilliantly turning Edson Alvarez on the left side of the area, only to be denied by Ochoa.
Philippe Coutinho then fired over goal and Ochoa saved from Gabriel Jesus, whose follow-up effort was cleared away by Jesus Gallardo.
Brazil maintained the pressure in the second half as Ochoa parried away Coutinho's shot.
"Everyone gives their best for the team. But things did not go well for Mexico today," said the goalkeeper.
He made 20 saves at this World Cup. Asked about his performances in Russia he added: “I am content about what I did at this World Cup but I can’t really be happy because we did not go through.”
He could do nothing moments later when Willian drove into the box and sent a low cross which Neymar poked home at the far post.
Paulinho should have doubled the lead when he was left alone near the penalty spot, but Ochoa once again came up with the save.
Mexico hit Brazil on a counter-attack that ended with Vela's shot being tipped over by Alisson, while Ochoa was on the spot again to save Willian's strike.
A bit of a flashpoint then occurred as Neymar collided with Miguel Layun, who then stepped on the forward's ankle as he went to retrieve the ball.
That caused Neymar to roll about theatrically on the ground, driving the ire of El Tri coach Juan Carlos Osorio, who felt the forward was being over-dramatic.
Brazil coach Tite defended his star in his post-match news conference.
“You saw it on the screen,” he said. “There's nothing to say.”
And it was Neymar who had the last laugh by beating an offside trap and sending a shot that was saved straight in the direction of Firmino, who tapped in from close range to settle matters.
Brazil will meet Belgium or Japan, in a quarter-final in Kazan on Friday.
It's the seventh consecutive quarter-final appearance for the five-time champions, while Mexico exit in the last 16 for the seventh straight World Cup.
“I would like to say once again that being able to play at such a level against a team like Brazil really shows Mexico is a good team,” coach Juan Carlos Osorio told reporters.
“We were not efficient and we didn't have that extra quality that they had.”
Both sides had early chances as Brazil keeper Alisson punched a cross straight to Hirving Lozano, whose shot was blocked by Miranda.
For Brazil, Neymar had an effort palmed away by the outstanding Guillermo Ochoa.
Lozano then just missed connecting with Javier Hernandez at the far post and had another shot blocked after a fine cross-field ball from Carlos Vela.
El Tri's confidence continued to grow as they worked down the left with Vela laying off for Hector Herrera, whose shot was snuffed out by Thiago Silva in the area.
Neymar then began Brazil's awakening by brilliantly turning Edson Alvarez on the left side of the area, only to be denied by Ochoa.
Philippe Coutinho then fired over goal and Ochoa saved from Gabriel Jesus, whose follow-up effort was cleared away by Jesus Gallardo.
Brazil maintained the pressure in the second half as Ochoa parried away Coutinho's shot.
"Everyone gives their best for the team. But things did not go well for Mexico today," said the goalkeeper.
He made 20 saves at this World Cup. Asked about his performances in Russia he added: “I am content about what I did at this World Cup but I can’t really be happy because we did not go through.”
He could do nothing moments later when Willian drove into the box and sent a low cross which Neymar poked home at the far post.
Paulinho should have doubled the lead when he was left alone near the penalty spot, but Ochoa once again came up with the save.
Mexico hit Brazil on a counter-attack that ended with Vela's shot being tipped over by Alisson, while Ochoa was on the spot again to save Willian's strike.
A bit of a flashpoint then occurred as Neymar collided with Miguel Layun, who then stepped on the forward's ankle as he went to retrieve the ball.
That caused Neymar to roll about theatrically on the ground, driving the ire of El Tri coach Juan Carlos Osorio, who felt the forward was being over-dramatic.
Brazil coach Tite defended his star in his post-match news conference.
“You saw it on the screen,” he said. “There's nothing to say.”
And it was Neymar who had the last laugh by beating an offside trap and sending a shot that was saved straight in the direction of Firmino, who tapped in from close range to settle matters.
