The 22-year-old Rodriguez struck on both halves of the match to not only sent Uruguay home but also emerged as the leading goalscorer of the World Cup so far with five goals in four matches. (Agencies)
Rodriguez notched up one of the most magnificent goal of the World Cup in the 28th minute when he took the ball on his chest just outside the penalty area and spun in a flash before smashing it left-footed over the fingertips of Muslera.
Colombia had the majority of ball possession initially and even though Uruguay, sans banned Luis Suarez, tried to dominate at the fag-end of the first half, the two-time winners could not get past the rival defensive line.
Rodríguez further piled up their agony when he doubled the lead in the 50th minute. Working the ball from the right, Pablo Armero feed Cuadrado, who headed the ball across the face of goal, where Rodriguez smashed it home.
Earlier, Hosts Brazil squeezed into the quarterfinals of the FIFA World Cup after a pulsating 3-2 penalty shootout win over heartbroken Chile at Belo Horizonte.
Brazil goalkeeper Julio Cesar saved Mauricio Pinilla and Alexis Sanchez's attempts, while Gonzalo Jara's shot hit the post after the prequarter-final had been locked at 1-1 after extra time.
For five-time winners Brazil, David Luiz, Marcelo and Neymar were the scorers, while Willian and Hulk missed out in the penalty shootout.
Chile, who were bidding to reach the quarter-finals for the first time since hosting the tournament in 1962, could have won the match in the last minute of extra-time but Pinilla's shot crashed off the crossbar.
Brazil too could have gone ahead in the 55th minute when Hulk had shot home a long ball from the left in the penalty area but English referee Howard Webb disallowed the goal and rather booked Hulk for handball.
The 22-year-old Rodriguez struck on both halves of the match to not only sent Uruguay home but also emerged as the leading goalscorer of the World Cup so far with five goals in four matches.