Uruguay finished as group winners after claiming all three matches, with Panama beating Bolivia 3-1 to finish second on six points, three ahead of the US.
The US needed a win in Missouri on Monday (Tuesday AEST) to have the best chance of progression. They started well, but the South Americans grew into the game as the first half wore on and went close to scoring through Darwin Nunez.
The biggest cheer of the night from the crowd at Arrowhead Stadium came in the second half when Bolivia scored an equaliser against Panama in the other Group C match, briefly moving the US up to second place, but their joy was short-lived.
Uruguay took the lead when Olivera bundled in a rebound after US goalkeeper Matthew Turner could only parry Ronald Araujo's header back into the box. The goal was given after a lengthy VAR review for offside.
Panama's Eduardo Guerrero (second right) scored his side's second goal in the 3-1 win over Bolivia. (AP PHOTO)
In Orlando, Florida, Panama beat Bolivia 3-1 to finish second in Group C and qualify for the quarter-finals.
Panama are playing in only their second Copa America and the victory moved them up to second with six points, three points above the US, as the central American side reached the knockout stage for the first time.
Uruguay finished with nine points.
Bolivia needed a win with only a slim chance of staying alive in the tournament, but it was Panama who drew first blood when Jose Fajardo expertly controlled a header into the box, let it bounce and volleyed home in the 22nd minute.
Bolivia equalised in the 69th minute when they cut through Panama's midfield and defence, with Ramiro Vaca finding Bruno Miranda, who slipped his shot past the keeper.
But Panama wrested back control 10 minutes later when Eric Davis curled in a cross from the left and Eduardo Guerrero connected with a glancing header.
Panama scored again in added time when Cesar Yanis was put through on goal and he volleyed the ball into the top corner to give them a two-goal cushion, sparking wild celebrations at the Inter&Co Stadium.
