Koji Miyoshi gave the tournament guests the lead in the 25th minute with a confident finish following a counterattack but Luis Suarez levelled seven minutes later from the penalty spot after a controversial decision following a VAR review.
Japan defender Naomichi Ueda, who appeared to be kicked by Edinson Cavani in a dispute for the ball, was penalised to the bemusement of the Asian side.
They restored their lead in the 59th minute when Miyoshi pounced on an error by Uruguay keeper Fernando Muslera to score his second of the game.
The 15-time Copa America champions equalised again seven minutes later through a header from defender Jose Gimenez.
Uruguay lead Group C with four points, while Japan are third with one.
Second-placed Chile meet bottom side Ecuador on Friday evening before the final round of Group C matches take place on Monday 24 June.