It was a solid start for Jose Pekerman's side whose preparation had been hit by injury concerns including the failure of Falcao to recover from a knee problem.
But Greece missed a number of chances to haul themselves back in the game and could have reduced the deficit on the hour when Theofanis Gekas headed against the bar from point-blank range.
Given the Greeks' reputation for frustrating their opponents with defensive tactics, Colombia's early breakthrough was celebrated wildly by the vast majority of supporters in the Estadio Mineirao.
Armero met Juan Cuadrado's low cross with a right-foot drive which took a slight deflection off Greek defender Kostas Manolas on its way into the net.
But the South Americans almost threw away their precious advantage immediately, Georgios Samaras squeezing a pass to the unmarked Panagiotis Kone in the box who fired a right-foot shot inches wide.
Colombian keeper David Ospina had almost nothing to do in the opening half-hour but Greece gradually gained a foot-hold in the game and Vasilis Torosidis flashed a header just wide from a Jose Holebas free-kick.
Rodriguez and Cuadrado spurned half-chances for Colombia before Kone had his side's best chance on the stroke of half-time, firing a rising right-foot effort which was superbly saved by Ospina.
Greece coach Fernando Santos sent his side out for the second half with more urgency, but despite the best efforts of Samaras it was the Colombians who eventually extended their lead.
A corner from the right was flicked across the face of goal by Abel Aguilar and Gutierrez stuck out a leg to put his side 2-0 up.
Gekas missed his golden chance to reduce the deficit on 61 minutes when his diving header from Kone's deep cross struck the bar and rebounded to safety.
As the pressure began to ease on the Colombians, they looked to take advantage of the counter-attack with substitute Santiago Arias finishing a mazy run by drilling a low shot at Kamezis on 76 minutes.
And Rodriguez grabbed the goal he deserved in injury time when he picked up a neat lay-off from Cuadrado to slide the third past the hapless Kamezis.
The result underlines Colombia's pre-tournament status as favourites to progress from Group C while Greece must look to get something out of their clash with Japan next Thursday in order to avoid continuing their record of never having progressed beyond the group stages of a World Cup finals.