The Argentines nearly took the lead on 19 minutes when another cross from Di Maria rebounded off the shins of Lionel Messi before falling into the path of Sergio Aguero, before he went to ground under the challenge of Santiago Arias and Colombia were able to clear the danger.
However, the World Cup-finalists' best chance of scoring before half time came when David Ospina was forced to make two saves from Aguero and Messi in quick succession.
Colombia almost gifted Argentina the lead moments before the interval when Cristian Zapata's block from Di Maria's cross nearly caught Ospina out.
Los Cafeteros continued to sit back after the break, but they almost snatched the lead midway through the half when Jeison Murillo hooked a corner towards goal, which Romero clung on to.
Substitute Carlos Tevez then almost won Argentina the game in the last couple of minutes as he nearly placed a shot past Ospina. However, Murillo recovered and scrambled the ball off the line.
That sent the game to penalties, and Colombia got off to a good start as James Rodriguez scored the first of the night.
Both teams rattled off a series of confident spot kicks, until Colombia's Luis Muriel missed the seventh one as he blazed it over the bar.
This gave Lucas Biglia the chance to win Argentina the game, but he skewed it wide, before Juan Zuniga missed the first in sudden death.
Marcos Rojo, also with the chance to win the game, then became the second Argentine to miss with the match on the line, before Murillo made it four misses in a row with an effort over the bar.
Tevez finally sealed the deal with a coolly-taken effort.