The victory sent Argentina to joint top of Group A with Chile on four points while Uruguay remain pointless.
These two sides played the first-ever international match outside the United Kingdom and have played each other more times than any other local rivals.
They've won the Copa 29 times between them but familiarity can breed caution and for much of the encounter they cancelled each other out.
Six-time Balon d'Or winner Lionel Messi had the first real chance, cutting in from the right onto his favoured left foot to force a full-stretch save from goalkeeper Fernando Muslera, who was relieved to see Lautaro Martinez unable to turn in the rebound.
Moments later, Cristian Romero rose highest in the area to meet a Rodrigo De Paul cross but Muslera made a comfortable save.
The pressure was relentless in the early stages and Argentina took a deserved lead on 13 minutes as Messi got to the byline and stood up a cross to the back post where Rodriguez rose above his marker to head home off the far post.
Uruguay gradually stemmed the tide and started to get a foothold in the game, only to see Argentina counter at pace before Muslera parried a fierce Nahuel Molina drive at his near post.
Despite enjoying the lion's share of possession, Uruguay were unable to test Argentina goalkeeper Emiliano Martinez.
A slew of early substitutions in the second half disrupted the flow of the game and chances were at a premium.
Despite boasting Luis Suarez and Edinson Cavani up front, Uruguay struggled to create any chances of note.
When Matias Vina put in a devil of a ball across the six-yard box, neither Suarez nor Cavani could get a touch to turn the ball in.
Soon afterwards, Suarez went for the spectacular with a leaping bicycle kick but sent the ball high over the bar.
Up the other end, Messi fired a free kick from 20 yards straight into the wall as the game petered out into a scrappy affair.