Emiliano Martínez emerged as the hero of the shootout, saving three penalties with the game ending 1-1 after extra time.
Argentina took an early lead when Lionel Messi, bidding to win his first major international competition, set up Lautaro Martínez to score from inside the Colombia box.
But Colombia equalized just after the hour mark when Luis Díaz gathered a quickly-taken free kick and lifted a shot past Martínez.
That was after both sides had chances in the first half — Wilmar Barrios hitting the post for Colombia and Yerry Mina hitting the bar from a corner, while Nicolás González was also denied by David Ospina at the other end.
Argentina had a golden chance to grab a winner midway through the second half when Ángel Di María rounded Ospina to set up Martínez, only for the striker’s shot to be cleared off the line by Barrios.
In the shootout, Juan Cuadrado and Messi both scored the opening penalties before Davinson Sánchez saw his effort saved by Martínez and Rodrigo De Paul launched his over the bar.
Martínez, who made a point of speaking to the Colombian players as they prepared to take their spot kicks, then saved Mina’s penalty.
Leandro Paredes netted his for Argentina and Miguel Borja responded by blasting his down the middle of the goal.
Goalscorer Martínez scored Argentina’s third from the spot, before goalkeeper Martínez produced his third save of the shootout to deny Edwin Cardona and hand his side victory.
“We’ve been shut in for 40 days, we were the only national team that couldn’t see anyone, we were really in our bubble by ourselves, with the coaches, staff, the cleaners, cooks and directors.
“It’s the work of a group of 70 people who came for a dream and we said it since day one: we want to play the final. And what better than playing Brazil on their pitch.”
Argentina will now face Brazil in Rio de Janeiro’s Maracana stadium this weekend in a bid to win their first Copa América title since 1993, while Brazil — which beat Peru in the semifinals earlier this week — is aiming for its sixth title since 1997.