The Monaco star scored arguably the goal of the tournament so far as Oscar Tabarez's men were dumped out of the competition
His unstoppable 28th-minute strike on the turn is arguably the goal of the tournament so far, and he followed that up with a close range effort in the second period to end the game as a contest.
Colombia dominated possession in a nervy opening, although they were mainly limited to shots from distance against a Uruguay side without star man Luis Suarez on the back of his four-month suspension.
Oscar Tabarez's men appeared content to sit deep but their resistance was broken when Rodriguez produced a spontaneous moment of brilliance to light up the match.
A lofted header back towards the Celeste penalty area was chested down on the turn by the Monaco man, who then swivelled and struck the ball first time on the volley into the top left-hand corner.
Stunned into action - with the rest of the stadium still taking in that wonder strike - Uruguay sought a swift response and Edinson Cavani found space on the left before his teasing cross was headed away by Carlos Sanchez.
Despite the improvement, Uruguay's task became even more difficult in the 50th minute as Rodriguez doubled his tally.
Colombia worked the ball from right to left and Pablo Armero put in a wonderful cross to the back post, where Juan Cuadrado unselfishly nodded the ball down and back into the path of the youngster, who swept home from close range for his fifth of the tournament.
Uruguay desperately needed a quick goal and Cristian Rodriguez forced a fine stop from Ospina with an accurate left-footed shot that would have found the top corner but for the goalkeeper's intervention, before Cavani flicked a header just over the crossbar in the 70th minute.
The Paris Saint-Germain forward went close again with a low-angled drive late on but Uruguay's pressure was comfortably soaked up, as Colombia stormed into the quarter-finals.