Full-back Pablo Armero had given the Colombians -- appearing at their first finals since 1998 -- the lead in the fifth minute while Teofilo Gutierrez's 13th goal for his country shortly before the hour mark wrapped up the three points.
Rodriguez, nicknamed the 'new kid' as he is seen as the successor to legend Carlos Valderrama, known as 'the kid', rounded off an impressive performance with a sweet left footed shot in time added on.
Armero's goal was the fastest in Colombia's five appearances at a World Cup finals, beating the 19 minutes it took another defender Francisco Zuluaga to score against Uruguay in the 1962 finals in Chile
It took Colombia just five minutes to send their fans wild.
With Falcao watching from the stands, great work down the right by Juan Cuadrado outwitted the Greek defence and a step over by Gutierrez saw the ball come to Armero.
The 27-year-old fullback fired in a weak shot which took a deflection off a defender and past 'keeper Orestis Karnezis.
The goal sparked an energetic goalline celebratory dance routine which drew in the players on the bench as well.
Greece, whose leading striker Kostas Mitroglou was left on the bench as he battles for full fitness after suffering a knee injury, had little to offer up front.
However, one bustling run from Celtic's Georgios Samaras set up Panagiotis Kone but he had to stretch for the ball and his effort curled harmlessly wide.
Kone made a far better fist of it right at the end of the first-half as once again Samaras's physical approach unnerved the Colombian defence and laid the ball back to his team-mate.
The 26-year-old fired in a powerful shot towards the top right hand corner but Colombian goalkeeper David Ospina produced a superb two handed diving save to deny him.
If the Greeks needed any encouragement to rescue the situation the fans provided them with it wielding a banner which read 'We won't say Greeks play like heroes, we say heroes play like Greeks'.
However, apart from one vain call for a penalty by Samaras the Greeks were incapable of threatening the Colombian goal and it was the South Americans who deservedly doubled their lead through Gutierrez.
Rodriguez's corner was a poor one but a superb deft flick on by Abel Aguilar came to Gutierrez who tapped home from close range -- his 13th goal in 31 appearances -- and provoked another superb dance celebration routine.
Colombia may have thought they were safe but the Greeks gave them a warning that all was not done as Vasilios Torosidis found the unmarked Theofanis Gekas yards out from the goal, but the 34-year-old's header beat Ospina only to cannon back off the bar.
It was the Colombians who rounded off the scoring as a flowing move ended with Monaco star Rodriguez side-footing the ball past Karnezis although the dance routine failed to materialise this time.
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