Jefferson Lerma put Colombia ahead after seven minutes when he took advantage of poor marking to steal in at the far post and head home a deep cross from the right.
But Falcao was on hand in a packed penalty box to clip a shot from Johan Mojica into the net and give Colombia a deserved share of the points.
Chile, who conceded a late goal against Uruguay to lose their opener 2-1, are on one point alongside Peru in the 10-team South American group, ahead of only Venezuela and Bolivia.
The next round of matches takes place from Nov. 12-17, with Chile at home to Peru and then away in Venezuela. Colombia host Uruguay and then travel to face Ecuador.