Martinez, the tournament's leading scorer, latched onto a precise through ball from Giovani Lo Celso before lifting his effort over advancing Colombia goalkeeper Camilo Vargas in the 112nd minute to spark celebrations among Argentina fans.
The Argentine national team star Lautaro Martinez was the tournament's top scorer, scoring 5 goals.
Argentina are the most successful team in Copa America, having won 16 titles. Uruguay is next best, winning the trophy 15 times.