It was the first time in his international career the FC Porto striker had scored more than once in a match, and the rest of the game, which was over as a contest, was as much about his quest to complete his hat-trick as anything else.
He failed, but Colombia, who kept their seventh clean sheet in their last ten matches, secured top spot in Group A, sending Bolivia out and leaving hosts Argentina needing a win over Costa Rica to secure second place.
Colombia coach Hernan Dario Gomez said his side had stuck to their gameplan and been rewarded. "We put them under early pressure, went out for the early goal, and we got it," Gomez noted with satisfaction.
Hugo Rodallega was also pleased: "We are extremely satisfied with how the group phase has gone. We came to Argentina with a goal [to qualify from the group] and we have done just that."
Falcao puts Colombia in front
Falcao opened the scoring in the 15th minute as he latched onto a straight pass from Dayro Moreno, his diagonal run taking him into the box, where he slotted the ball past Carlos Erwin Arias.
Bolivia paid for their lack of adventure just before the half-hour as they handed their opponents a second. A midfield mix-up allowed Pablo Armero to surge into the left side of the penalty area, where he was scythed down by Santos Amador.
It was a clear-cut penalty and Falcao made the most of it, his stuttering run-up causing Arias to move early as he sent the ball down the middle.
Gustavo Ramos should have added a third before the break but he contrived to miss from close range after good work from Camilo Zuniga and Fredy Guarin.
Fails to hit hat-trick
The second half continued in a similar vain with Colombia dominating, playing neat football without creating many good chances.
Falcao went for his hat-trick after 64 minutes when substitute Rodallega laid the ball off, but the ball flashed wide. Then a minute later Moreno chose to shoot, missing the target, when Falcao was lurking.
After 72 minutes Colombia came mightily close to adding a third when an intricate move involving Gustavo and Armero ended with Rodallega shooting for goal. Falcao, desperate for his hat-trick, stuck out a foot and deflected the ball wide.
Falcao had another chance for a third six minutes from time when Juan Cuadrado's cross found him at the far post, but he could not jump high enough and headed over.