Their plan nearly worked early on as Argentin keeper, Franco Armani, played a risky pass to Nicolas Otamendi who miscontrolled it and almost led to an opening goal.
Colombia initially looked stronger but Argentina were a different beast when Rodrigo de Paul replaced Angel di Maria immediately after the interval and captain Lionel Messi was able to find space to cause some problems.
Argentina lost the final to Chile in the last two editions of the competition but Messi insisted the team shouldn't be considered one of the favorites this time as the squad has plenty of young players entering their first major global tournament.
"In the first half, even though they had a lot of the ball, they didn't generate chances. Now we have to think about Paraguay".
La Albiceleste are bottom of the table after being beaten 2-0 by Colombia first up, while their opponents opened with a 2-2 draw versus Qatar.
Messi briefly came to life, nutmegging Davinson Sanchez to burst into the area, but the tireless Wilmar Barrios got back to crowd out the Barcelona superstar.
What had been a petulant affair threatened to boil over when Cuadrado was booked for a foul on Messi that sparked a brief melee of pushing and shoving.
Argentina and Colombia promise fireworks in a game featuring wonderful individual talents and two teams who looked really good in their warm-up friendlies.
But against the run of play, Colombia struck a hammer blow to Argentina and Messi's Copa hopes.
"It's not the best way to start, but if we keep lamenting and don't move on, if we don't pick ourselves up, we won't be prepared to face Paraguay in the next match", said the 31-year-old Messi, Argentina's oldest player in the Copa America squad.
Substitute Martinez cut in from the left and drove his effort into the top corner in the 71st minute after being set up by James Rodriguez.
It was more of the same in the second half, though Colombia goalkeeper David Ospina was finally called into action after a vicious volley from Leandro Paredes approaching the hour mark. Zapata was on the pitch just five minutes when he directed Jefferson Lerma's cross over the line.