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.
A penalty from Arturo Vidal after 38 minutes and an opportunist goal from Alexis Sanchez three minutes before halftime then put Chile in the driving seat.
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.
