After having a day off at the weekend when the opening round was played, the Cafeteros hit their straps in La Paz, some 3,600metres above sea level.
Dorlan Pabon put Colombia ahead two minutes after the interval, but Walter Flores equalised with a long-distance shot with only five minutes to go.
Bolivia were close to their first point - after a 4-2 reverse at Uruguay - but Falcao finished a counter-attacking play to give Colombia the three points.
Colombia boss Leonel Alvarez told a media conference: "I want to see how many teams will come to La Paz top lay the way Colombia did. This is a very difficult place."
Bolivia coach Gustavo Quinteros said: "Colombia played much better.
"We did not play as we had planned. We were not solid at all."
Chile overcame a disappointing 4-1 defeat at Argentina at the weekend as they claimed a 4-2 triumph over Peru in Santiago.
Claudio Borghi's Chile needed only one minute to take the lead as Waldo Ponce sent them in front after a Jorge Valdivia corner.
In the 18th minute, Jean Beausejour doubled the lead before a 25-metre shot from Gary Medel made it 3-0 two minutes after the interval.
Peru reduced the deficit with Claudio Pizarro and Jefferson Farfan scoring at the 49th and 59th minutes, but Humberto Suazo consolidated Chile's win four minutes later.
Playmaker Valdivia told the media: "It was a clear difference in the first half. The three points are very important.
"In the second half, Peru scored two goals quickly, but we managed to net a fourth to calm us down."
Finally in Asuncion, Paraguay gained one point in their home duel with Uruguay as they drew 1-1.
Reigning Copa America champions Uruguay took the lead in the 68th minute, when Diego Forlan was assisted by Luis Suarez and beat Paraguay keeper Edgar Barreto.
However, the Guaranies never gave up and Richard Ortiz leveled the duel in the second minute of injury time.
Later in the day, Argentina slumped to a shock 1-0 defeat to Venezuela.
Athletic Bilbao defender Fernando Amorebieta etched his name into the history books by scoring the only goal of the game in the 61st minute to lead Venezuela to their first ever win over the Albiceleste.