Rodriguez, making his first start for his country since June 2019, headed in off a post early on following a cross from Lionel Messi.
Uruguay had a penalty appeal by striker Edinson Cavani turned down as they failed to register a shot on target in their opening match of the tournament.
Argentina’s win moved them joint top of their group with Chile on four points.
“We deserved to win,” said Argentina midfielder Rodrigo de Paul.
“We tried to be more compact, to make it more difficult to score against. Up front we have monstrous players who can get goals at any moment of the match.”
Uruguay’s next game is against Chile (21 June), while Argentina are next in action against Paraguay (22 June).