With Falcao watching on from the stands, his Monaco teammate James Rodriguez the star man as after Pablo Armero's early opener, he set up Teofilo Gutierrez to double the South Americans' advantage before rounding off the victory himself in stoppage time.
Victory propelled Colombia to the top of Group C with Japan and Ivory Coast to meet later in the day.
"We are capable of scoring goals and creating chances, which is a great thing to have," said Pekerman, who coached Argentina at the 2006 World Cup.
"The important thing is that everyone did their bit for the team and that transmits confidence. We took our chances and knew how to play as a team."
"We have maintained the identity of the team and backed it up with defensive work. Greece have very experienced players and have played in many World Cups. It was a very positive game.
"It am very happy with the win and the margin of victory. That is very important to allow us to progress in the tournament."
Colombia are appearing in the finals for the first time since 1998, but much had been expected of Pekerman's men due to an impressive qualifying campaign before star man Falcao was sidelined by a cruciate ligament injury.
However, Rodriguez, assumed the pressure placed upon him in Falcao's absence with a man-of-the-match display that he hopes is just the beginning of a fine tournament.
"I am happy because Colombia won and I also scored a goal, that gives me great joy. I hope we can continue to win," said the 22-year-old.
Next up for Colombia is a clash against the Ivorians on Thursday.
A win in Brasilia would almost certainly guarantee their place in the last 16 where they could meet former World Cup winners Italy, England or Uruguay.