The South Americans made a blistering start with left back Pablo Amero scoring with a 5th minute deflected shot. Striker Teofilo Gutierrez extended their lead, stabbing home a 58th minute corner, and midfield James Rodriguez added a late third goal to seal the win.
The Group C opener in Belo Horizonte's Estadio Mineirao showed that Colombia remain a force to be reckoned with despite the injury to Falcao, which overshadowed their buildup to the tournament.
Greece had plenty of chances themselves.
In their best opportunity, veteran striker Fanis Gekas hit the bar with a header when it looked easier to score soon after Colombia's second goal.
Vassilis Torosidis put a header just wide, Panagiotis Kone saw a shot well saved and Giorgos Samaras struck just wide for the normally defence-minded Greeks, who were forced to attack after Colombia's early goal.
The result pumped up an already pulsating atmosphere among the majority Colombian crowd. The yellow-decked South American fans were thrilled at seeing their team back at football's top table after being shut out since their great sides of 1990s.
That joy was in evidence after Colombia's first goal, when the players danced together in probably the most joyous celebration so far of the Brazil World Cup.
Defeat continued Greece's dismal record of never keeping a clean sheet at a World Cup finals. They have never gone beyond the group stage either, so will be hoping Group C rivals Ivory Coast and Japan prove easier opponents.
Armero's early goal came after sustained pressure down the right wing, with a lucky deflection squeezing past keeper Orestis Karnezis. Three Colombian players looked offside but were deemed not interfering in play.
Gutierrez, with a heavy weight of expectations on his shoulders given Falcao's absence, scored an easy goal after a corner had been flicked on.
Then Rodriguez, who had looked lively throughout, hit a low shot in stoppage time to send the Colombian fans delirious.
Throughout the game, the Colombians targeted Greek left back Jose Holebas, again and again finding space on the right flank.
Greek forward Kostas Mitroglou came off the bench in the second half but was unable to provide Greece's missing spark.