Chile blitzed the home side early to lead 3-0 before the half-hour mark with Barcelona forward Alexis Sanchez central to the visitors' start as he scored his team's second and third goals.
But a red card to Chilean midfielder Carlos Carmona in the 66th minute allowed Colombia to recover with Teofilo Gutierrez scoring the home side's first three minutes later before James Rodriguez was fouled in the box on two occasions and, both times, Falcao converted from the spot.
Meanwhile, Chile missed the opportunity to qualify and will need to claim at least a point at home against Ecuador on the final matchday in South American qualifying to secure a berth at next year's tournament.
Just three minutes later and Jorge Valdivia set up Sanchez for Chile's second and the visitors were flying in the 29th minute when the Barcelona striker struck again, finishing off Marcos Gonzalez's pass.
Soon after Gutierrez struck from Pablo Armero's cut-back and in the 73rd minute Rodriguez went down easily under pressure from Francisco Silva and Falcao scored to make it 3-2.
Colombia's comeback was complete in the 84th minute when Rodriguez was brought down by Chilean goalkeeper Claudio Bravo and Falcao again made no mistake from the spot.
