In an encounter that saw both sides reduced to 10 men, Atletico Madrid striker Falcao scored one and set up another as Colombia remained in second place in the table.
Despite midfielder Gary Medel being sent off in the 34th minute for hitting Luis Perea, Chile took a half-time lead through Matias Fernandez following a blunder from the Colombian defence.
In the 41st minute, the Colombian players stopped running as they expected referee Victor Carrillo to blow a foul on Edwin Valencia.
They left Matias Fernandez unmarked and the Chilean forward's strike from outside the box beat Colombia keeper David Ospina.
However, James Rodriguez levelled just before the hour mark with a free-kick, while Aldo Ramirez teed up Falcao, who put Colombia ahead in the 73rd minute at the Estadio Monumental.
Two minutes later, Teofilo Gutierrez linked up with Falcao before finishing with aplomb although Colombia were also then reduced to 10 men when defender Abel Aguilar was sent off for a second bookable offence with six minutes remaining.
"This is a magnificent win for us. We were sure it was going to happen," Falcao told reporters in Santiago.
"This is the result of the maturity of a team who did not give up after Chile's goal."
Colombia are now on 13 points in the CONMEBOL qualifying zone, the same as Ecuador, who played out a 1-1 draw with Uruguay in Montevideo.
Former Manchester City striker Felipe Caicedo put Ecuador ahead with a seventh minute penalty before Edinson Cavani gave the Uruguayans a share of the spoils in the 66th minute.
However, the hosts were a perhaps a touch fortunate after keeper Fernando Muslera appeared to foul Cristian Benitez in the area.
Referee Carlos Amarilla refused to award a penalty and Ecuador then saw Manchester United midfielder Antonio Valencia sent off in injury-time.
Argentina remain top of the group despite a 1-1 draw at Peru in Lima.
Carlos Zambrano had put the hosts in front after 22 minutes but Real Madrid striker Gonzalo Higuain (38) drew level for Argentina.
Salomon Rondon's double gave Venezuela a shock 2-0 victory at Paraguay.
Rondon struck in either half as Venezuela moved into sixth in the table on 11 points, just one below Chile, who occupy the last automatic qualifying position at the moment.