Chelsea made most of the running and dominated early on but were reckless with their finishing and went into the break with a one-goal advantage thanks to Cole Palmer's 16th-minute strike on Friday.
Palmeiras roared back after halftime with teenage winger Estevao, who is soon to join Chelsea, equalising in the 53rd minute with a stunning strike from a tight angle.
Seven minutes from time, Gusto's attempted cross deflected off defender Bruno Fuchs and wrongfooted Weverton, the ball bouncing off his back and into the back of the net.
Chelsea will now face Fluminense in the semi-finals in New York on Tuesday, after the Brazilians continued their fairy-tale run with a 2-1 victory over Saudi Arabia's Al-Hilal in Orlando.
The tournament underdogs struck first through Matheus Martinelli in the opening half before Al-Hilal hit back after the break when Marcus Leonardo found the net.
But Fluminense refused to be denied and regained their lead in the 70th minute through Hercules to secure a memorable win over Al-Hilal in the first meeting between the clubs.
The Brazilian side, who entered the tournament as one of the biggest long shots, will now face the winners of Friday's other quarter-final clash between Palmeiras and Chelsea.
Al-Hilal announced themselves in the footballing world when they beat Manchester City 4-3.
Liverpool FC and Portugal national team star Diogo Joto and his brother Andre Silva, who died Thursday in Spain in a car accident, were honoured with a pre-game moment of silence.
Cameras showed Al Hilal starters and Portuguese compatriots Ruben Neves and Joao Cancelo fighting off tears during the observance.
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