The five-time Ballon d'Or champion also set up Angel di Maria with a tap-in six minutes from time.
Messi opened the scoring for La Albiceleste in the 10th minute with a glorious, bending free kick that left Colombia's David Ospina no chance to make it 1-0.
And the hosts were at it again in the 23rd minute, Messi's whipped-in cross finding Lucas Pratto, who flicked his headed attempt back across the goalmouth and in at the far post.
The extremely physical encounter continued throughout the second half, with both teams guilty of hard fouls and eight yellow cards handed out in all.
Colombia pushed hard for a goal, but it was Argentina that would strike again, Messi turning Colombia over in their defensive third, dribbling into the penalty area and finding Angel Di Maria at the far post for an easy smash into Ospina's net.
The win sees Argentina leapfrog Colombia in the qualification table.