Jose Pekerman's men, making a first appearance at the finals of the World Cup since 1998, produced a strong showing to enhance their hopes of qualifying for the last 16.
Star striker Radamel Falcao's omission from the squad had dominated the build-up to the tournament for Colombia.
However, backed by a vast and vociferous travelling support at the Estadio Mineirao in Belo Horizonte, Colombia showed that they could perform without the Monaco forward.
Defender Pablo Armero opened the scoring early on, before Teofilo Gutierrez - the man tasked with leading the line in the absence of Falcao - doubled his side's advantage just before the hour mark.
James Rodriguez added a third in injury time to put the seal on victory for Colombia in Group C's opening game.
Colombia's next match comes against Ivory Coast on Thursday, while Greece will bid to bounce back on the same day versus Japan.
The South American outfit made the most of a fast start to proceedings, moving ahead after just five minutes in somewhat fortuitous circumstances.
Armero latched on to a low right-wing cross from Juan Cuadrado, and his seemingly tame effort deflected off Greece defender Kostas Manolas, deceiving goalkeeper Orestis Karnezis and nestling in the bottom-left corner.
Greece could have levelled immediately if not for the disappointing finishing of Panagiotis Kone, who fired wide from the edge of the area.
Colombia, with their pacy and exciting forward line, continued to offer more of a threat, although Greece spurned another chance to level in the 28th minute as Vasilis Torosidis headed off target from a free-kick.
Fernando Santos' Greece began to dictate matters towards the end of the first half and were unfortunate not to equalise when Kone forced a fine save from David Ospina with a curling 20-yard effort.
However, Pekerman was evidently able to re-energise his side during the half-time interval and Colombia pushed for a second goal after the break.
They were rewarded for their industry in the 58th minute, Gutierrez turning the ball into an unguarded net after Abel Aguilar had flicked Rodriguez's near-post corner into his path.
Fortune favoured Colombia throughout and Greece's bad luck was encapsulated after 63 minutes, striker Theofanis Gekas rattling the crossbar when he met Torisidis' knock back from a deep far-post cross with a powerful diving header.
Gekas was immediately replaced by Kostas Mitroglou, but the change had little effect.
And Colombia duly put the gloss on a fine win in injury time as the influential Rodriguez sent a cool finish beyond a despairing Karnezis.
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