Argentina were off to a flying start, helped by woeful home defending, to move 2-0 up inside eight minutes through goals by Cristian Romero and Leandro Paredes.
But Colombia’s substitutes turned the game around in the second half as Luis Nuriel netted from the spot and then Borja snatched a point at the death.
Argentina remain unbeaten in second place while Colombia are just outside the automatic qualification spots in fifth.
The visitors will be kicking themselves, though, after a fast start and having the better chances throughout.
Three minutes in, Romero rose highest to head home a Rodrigo De Paul free-kick from wide on the right that made it all the way to the edge of the six yard box.
A few minutes later, Lionel Messi picked out Marcos Acuna and his rasping left-footed drive was deflected inches past the corner of post and crossbar.
Colombia failed to fully clear the resulting corner and following some pinball in the box, Paredes was able to run past four players, find space and shoot low into the corner across goalkeeper David Ospina.
Argentina were well on top and Ospina had to save a fierce drive from Lautaro Martinez on 27 minutes, with Nicolas Gonzalez putting the rebound a whisker past the post.
Colombia finally came alive in first half stoppage time but Muriel’s shot was blocked by Nicolas Otamendi, while Duvan Zapata blazed over.
They were back in it six minutes into the second period, though, after Otamendi brought down Mateus Uribe in the box and Muriel slotted in from the spot.
Argentina remained well on top, though, and Ospina tipped a shot from Messi onto the bar before denying Martinez.
Muriel shot over before the inspired Ospina kept Colombia in the tie with great saves from Martinez and Messi again.
Borja had the last word, though, rising at the back post to head home Juan Cuadrado’s cross.
Peru earned their first win of the qualification campaign but remained bottom of the group despite a surprise 2-1 success at Ecuador.
Christian Cueva converted a rapier counter attack on 62 minutes and Luis Advincula all but secured victory eight minutes from time, shooting through goalkeeper Alexander Dominguez’s legs.
Gonzalo Plata’s injury time strike was but a consolation for Ecuador, who nonetheless remain third.
In Caracas, Venezuela held Uruguay to a 0-0 draw to remain above Peru on goal difference, while the visitors clung on to a top four automatic qualification spot.