Colombia took seven points from their three matches, the best return of the tournament so far, to qualify for a quarter-final against the best third-placed team from the three groups in Cordoba next Saturday.
Bolivia were eliminated while hosts Argentina and Costa Rica are set to play for second place in Cordoba on Monday (0045 GMT Tuesday).
Falcao is now joint top scorer of a low-scoring tournament with a tally of two, equal with Peru's Paolo Guerrero, and the team went through without conceding a goal.
"These results put us in the quarterfinals with authority," Colombia's coach Hernan Dario Gomez told the post-match news conference.
"Our objective is not to go looking for a goalscorer but our identity as a team," he said after Falcao scored the goals but Colombia created chances for several different players.
Falcao put the Colombians ahead after a quarter of an hour when he ran onto winger Dayro Moreno's defense-splitting ball down the left to steer it past goalkeeper Carlos Arias.
Colombia went further head close to the half hour with a penalty converted by Falcao after defender Santos Amador brought down left back Pablo Armero.
It was the second penalty of the tournament and the second faced by Arias after he saved one from Costa Rica's Joel Campbell in Bolivia's 2-0 defeat in Jujuy on Thursday.
"With the attitude of going out to win all the matches we knew things would happen for us. Today we got first place but if we don't make the most of it in the next match, it stops there. We want to carry on dreaming," Falcao told reporters.
Midfielder Adrian Ramos almost made it three when his header from a perfect high cross from Armero went just over the bar.
Colombia's goalkeeper Luis Martinez had trouble turning away a free kick from midfielder Ronald Garcia near the end of the first half and Jhasmany Campos had a half chance for Bolivia early in the second but his effort went over.
Colombia created several chances to increases their lead with Falcao shooting wide from the best after being set up by substitute Hugo Rodallega, who later had a shot on target deflected by a defender.
