Alejandro Sabella's side controlled possession during most of the first half, but could not find a way through the Cafeteros' defence until Lionel Messi and Sergio Aguero were on target in the second half.
It was the hosts who broke the deadlock on the stroke of half-time, midfielder Dorlan Pabon's free-kick taking a wicked deflection off Javier Mascherano.
The visitors were on level just after the hour mark, though, thanks to an error by Colombia goalkeeper David Ospina. The Nice stopper spilled Jose Sosa's cross, allowing Barcelona forward Messi to flick the ball into the empty net.
Argentina sealed an unconvincing win five minutes from time on the counter-attack. Messi fed Real Madrid striker Gonzalo Higuain, whose cross-shot was saved by Ospina but Manchester City frontman Aguero was on hand to tap home the rebound.
Man of the match Messi said: "It was hard. We did not deserve to go one goal behind to the opponents, as they were not better than us.
"We wanted to forget about the tie against Bolivia [1-1 draw on Friday]. That was very hard for us and I think we did.
"Sometimes you can play better, and other times you have to fight harder. This game was one of those days."
The victory moved Sabella's team on to seven points from four games - level at the top of the South American zone standings with Uruguay, who have a game in hand.
Higuain added: "We wanted to win. We were very low after the game on Friday and we promised to do better here.
"It was a great match, very difficult to play at a demanding venue, but we improved our performance in the second half and did it.
"Now, we have six months to try to find our best shape. The qualifiers are long and we made a good step today."