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A win for either team would have guaranteed them a place in the last eight and Japan had the better of the early stages and took the lead in 15 minutes when Shoya Nakajima scored from 25 meters out.
Brazil's preparations for the Copa have been overshadowed by Neymar -- firstly the rape allegations against the striker, which he has denied, and then the ankle injury that ruled him out of the tournament last week.
Argentina's win, their first after a loss to Colombia and a draw with Paraguay, leaves them in second place in Group B on four points behind group winners Colombia, who beat Paraguay 1-0.
Suarez spurned the first penalty of the shootout, his effort bouncing off the chest of Peru goalkeeper Pedro Gallese, to complete a frustrating evening for Uruguay's all-time top scorer, who also had a goal ruled out for offside.
The Selecao's golden boy is dominating the build-up to a tournament he will miss due to an ankle injury for all the wrong reasons.
Messi and Argentina can still appeal the decision, which stops the striker from playing in four friendlies this year.
The 31-year-old made 75 appearances for Argentina, scoring 31 goals. His last appearance was in the 2-1 win over Nigeria in the 2018 World Cup.
The decision marks the first time in the modern Copa America era that the tournament will be split between two countries.
Results 101-108
