The Mexican giants were beaten 1-0 at the Estadio Azteca to miss out on a spot in the knockout stages from Group 4.
The only goal of the match came nine minutes after the break when Jerry Palacios nodded home a superb left-wing cross from Ariel Soto.
The Honduran striker escaped the attention of his defender, meeting Soto's inch-perfect cross with a powerful downward header from 10 yards out that gave the 'keeper no chance.
Despite some nervy moments in the final stages, the Costa Ricans survived some close calls to hold on for victory and leapfrog America into top spot in the group to progress to the last eight.
There was better news for a Mexican club in Tuesday's other match, with Cruz Azul confirming their spot in the quarter-finals with an away win over Herediano.
Only needing a draw to progress, Cruz Azul claimed maximum points with Jesus Lara Garcia's 88th-minute strike handing them a 2-1 win at the Alejandro Morera Soto in Group 3.
The visitors made the perfect start when Achille Emana gave them the lead inside five minutes after some brilliant lead-up play by Mariano Pavone.
But the Costa Rican club – who needed a big win – gave themselves hope when Anllel Porras equalised 10 minutes after the break.
Cruz Azul were largely happy to sit back and hold on for the draw they needed to progress before Garcia iced the result late on.
The victory means Cruz Azul move on to the last eight with a perfect record of four wins from four matches from the group stage.
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