It was Carlos Tevez who scored the winning penalty following a string of unbelievable misses at the Estadio Sausalito in Vina del Mar.
Argentina thoroughly deserved the victory as they were dominant throughout the 90 minutes. But, Colombia and Arsenal goalkeeper David Ospina was the reason why Argentina were kept at bay throughout the match and were made to work extremely hard to win the penalty shootout.
Ospina's highlight of the match came in the form of a breathtaking double save in the 26th minute when he initially denied Sergio Aguero from scoring with his foot before scrambling to palm away Lionel Messi's follow-up header.
Colombia coach Jose Pekerman made four changes to the team that drew 0-0 with Peru, but they proved to be largely ineffective as Argentina constantly sliced their way through Colombia's defence and came close to scoring on numerous occasions.
Ever Banega nearly put Argentina 1-0 up almost immediately after coming on for Javier Pastore in the 78th minute when his scorching shot from 25 yards ricocheted off the crossbar.
Argentina came agonisingly close to scoring again just two minutes later, but defender Nicolas Otamendi's volley from Angel Di Maria's corner was sensationally tipped onto the post by Ospina before it was cleared by Cristian Zapata.
With neither team able to break the deadlock when the final whistle was blown, penalties were needed to decide who would go through to the semi-finals.
Argentina and Colombia both converted their first three penalties before Colombian striker Luis Muriel blasted his shot some distance over the crossbar.
Argentina forward Ezequiel Lavezzi gave his side a 4-3 lead despite Ospina getting a touch on the ball before Edwin Cardona equalled things up.
Lucas Biglia failed to seal the deal for Argentina as he placed his penalty wide of the post before Juan Zuniga had his shot saved by Sergio Romero.
With Marcos Rojo hitting the crossbar and Jeison Murillo's attempt going high and wide, it was Tevez who produced the killer blow as he made no mistake with his penalty.
Argentina will now face Brazil or Paraguay in the semi-finals.