Hoffenheim forward Vargas completed a hat-trick in 13 devastating minutes either side of half-time to send Chile into a semi-final against Colombia in Chicago on Wednesday.
Vargas later dedicated his performance to his mother, who was earlier this week diagnosed with a heart problem in his homeland.
“I’m very happy with the performance, I dedicate it to my mum and my family,” Vargas said.
“We started this tournament badly but we’re getting stronger and stronger with every match,” he added.
The drubbing was Mexico’s worst ever defeat in the finals of a major tournament. Their previous biggest loss was a 6-0 rout by West Germany at the 1978 World Cup in Argentina.
The rout was also only one goal away from equalling Mexico’s heaviest ever defeat in any international, an 8-0 thrashing by England in London in 1961.
– ‘A disgrace’ –
“I offer my sincere apologies to the Mexican fans, to all the Mexican people,” Mexico coach Juan Carlos Osorio said afterwards.
“What we saw today was a disgrace, an accident of football,” he added. “Obviously there are more important things in the world than football, but that was a very poor display.
“Chile had a great collective performance and extraordinary performance from its stars,” Osorio added. “But there is not seven goals difference between Chilean and Mexican football.”
Vargas grabbed his fourth goal in the 74th minute to give Chile a 6-0 lead.
But the punishment did not finish there with Edson Puch adding a seventh in the 88th minute for his second of the match.
Chile, winners of last year’s Copa America on home soil, had faced an intimidating atmosphere in front of a crowd of 70,547 people at the Levi’s Stadium, home of the San Francisco 49ers.
But despite the overwhelmingly pro-Mexico crowd, Chile were quicker to settle, with Puch opening the scoring on 16 minutes when he steered in a rebound from Marcelo Diaz’s shot after Guillermo Ochoa could only parry away.
Vargas grabbed a second a minute before half-time to stun the crowd into silence as the teams headed in for the break.
Chile then cut loose early in the second half with slick inter-play ripping apart a feeble Mexican defense at will.
Arsenal star Alexis Sanchez made it 3-0 in the 49th minute before two more goals from Vargas in the 52nd and 57th minutes piled on the agony for Mexico.
Several Mexican players visibly wilted in the face of the onslaught, and for periods the game resembled a training exercise for Chile as they swept forward relentlessly.
Vargas’s fourth made it 6-0 on the night before Puch then toe-poked home to complete the scoring.
Brazilian referee Heber Lopes spared Mexico the possibility of greater humiliation, whistling for full-time almost on the stroke of 90 minutes, with no time added on.