A pulsating clash at the University of Phoenix stadium in Arizona saw Mexico take all three points in the crucial Group C battle through an own goal from Alvaro Pereira and late strikes from Rafa Marquez and Hector Herrera.
Uruguay's lone strike came from captain Diego Godin, the Atletico Madrid player heading in an equalizer 16 minutes from time that looked to have secured the South Americans a share of the points.
But 37-year-old Marquez's emphatic strike into the roof of the net on 85 minutes fired Mexico 2-1 up before Herrera added a third in injury time to complete a memorable victory for "El Tri".
Uruguay's players angrily remonstrated with match officials at the final whistle in an ill-tempered finale that capped a miserable afternoon for 'La Celeste'.
Before kick-off, Uruguay's players had looked baffled as they lined up for their national anthem – only for Chile's national anthem to be played by mistake.
Copa America Centenario organisers swiftly apologised for the blunder.
"We sincerely apologise to the Uruguayan Federation, the Uruguay National Team, the people of Uruguay and to the fans for this mistake," a statement said, blaming the gaffe on "human error."
When the game kicked off, Mexico took the lead after only four minutes with a goal that owed everything to the guile of PSV Eindhoven midfielder Andres Guardado.
The 29-year-old wide player carved out space for himself on the left flank superbly and then whipped in a wicked cross that sowed panic in the Uruguayan defense as Herrera lurked menacingly.
Alvaro Pereira spotted the danger but as he scrambled to neutralize the threat, succeeded only in powering a header into his own net.
