The Argentinians had striker Gonzalo Higuain sent off after a clash with Colombia's Cristian Zapata midway through the first half and also saw two second-half goals from Sergio Aguero disallowed for offside.
Argentina, who lead the standings by five points and remain firmly on course for an automatic qualifying berth with four games left to play, started a gripping contest at the Monumental Stadium with skipper Lionel Messi on the bench.
The Barcelona star, who is nursing himself back to fitness after a hamstring injury, was eventually brought on early in the second half but was unable to break the deadlock for Argentina against a fiercely committed Colombian outfit.
The point saw Colombia move up to second ahead of Ecuador on goal difference.
The frenetic pace saw players from both sides flying into some full-blooded challenges, and tensions boiled over on 26 minutes when Higuain and Zapata were given their marching orders.
Zapata saw red after appearing to lash out with a boot in retaliation for a sly kick by Higuain at the back of the defender's legs as he thwarted the Argentinian.
It was a sour note to a gripping first half, which had seen Argentina repeatedly threaten the Colombian goal early on only to be denied by some phenomenal saves from the visiting goalkeeper David Ospina.
Ospina needed to be alert on 13 minutes when Real Madrid star Angel Di Maria rifled in an angled volley from the edge of the area which the Colombian keeper gathered.
Moments later Colombia should have taken the lead when Napoli's Juan Zuniga burst down the right and crossed for Radamel Falcao.
Falcao, who recently signed for French side Monaco in a deal reported to be worth around €60 million, had time and space but could only screw his header wide from close-range.
Three minutes later and Argentina were left wondering how they had not taken the lead after a stunning double save from Ospina.
Higuain pounced on a slip to shoot from close range but Ospina blocked. The rebound fell to Marcos Rojo, but Ospina somehow managed to tip the Sporting Lisbon defender's header onto the bar as Colombia escaped.
Colombia had the ball in the net on 20 minutes but Jackson Martinez's effort was correctly adjudged to be offside.
The frenzied pace continued after the sendings off of Higuain and Zapata but not even the introduction of Messi could tip the contest Argentina's way.
Aguero's had the ball in the net on 54 minutes but was flagged for a fractional offside. And the Manchester City star was denied again in a tight decision on 72 minutes when he scrambled in from close range.
In Friday's early game, Venezuela were denied a precious victory against Bolivia in La Paz after Jhasmani Campos's late equaliser cancelled out Juan Arango's goal for the visitors as the game finished 1-1.
The result saw Venezuela inch ahead of Chile into fourth place in the standings with 16 points from 12 games. Chile however will reclaim fourth spot if they beat bottom side Paraguay in Asuncion in the late game.