Rodriguez steered the ball with the inside of his left foot into the bottom corner of Duwayne Kerr’s net in the 52nd minute after fellow midfielder Nicolas Lodeiro’s free kick on the left was headed back into the middle by right back Alvaro Pereira.
Uruguay missed the incisive play of suspended striker Luis Suarez, who inspired them to their title triumph in Argentina four years ago, and substitute Cristhian Stuani miskicked a good chance to score a second goal two minutes from time.
Jamaica tried to make their superior height pay in the final quarter of an hour as they pushed for an equaliser but striker Giles Barnes and substitute Deshorn Brown put headers wide.
The Caribbean islanders failed to really test goalkeeper Fernando Muslera despite plenty of possession.
"Today we had tough rivals, they chased every ball. It wasn’t our best afternoon but we have several days' rest to think about the next match,” Rodriguez told reporters.
"We got a positive result which is important... Now to think about Argentina as a final,” said captain Diego Godin ahead of Tuesday’s clash with Lionel Messi’s tournament favourites.
Jamaica were left to rue poor defending at the free kick that produced the goal and some disappointing finishing and set pieces despite taking the game to Uruguay for long spells.
It took until the 37th minute for a direct shot on goal with Kerr diving to his right to push away a Rodriguez effort.
Less than three minutes later Jamaica looked set to take a shock lead but striker Darren Mattocks, who had chested the ball past the centre backs, mishit in front of goal and Godin scrambled the ball clear.
Edinson Cavani, leading the Uruguay attack, had a frustrating match and remonstrated with Venezuelan referee Jose Argote at halftime over being denied a penalty when he appeared to be pulled back in the 41st minute.
Cavani was involved in Uruguay’s last meaningful attack that should have brought a goal in the 88th minute but Stuani's shot hit the ground in front of Kerr and bounced over the bar.
Image: Uruguay's Cristian Rodriguez (right) celebrates his goal with team mate Jose Gimenez.
Photograph: Andres Stapff/Reuters