After a goalless draw in which Argentina did everything but score against a Colombian side inspired by one of the tournament's great goalkeeping performances, Tevez stepped up to beat David Ospina and seal the shootout 5-4.
Argentina will play either Brazil or Paraguay in next week's semi-finals.
Tevez, who finished the job after team mates Lucas Biglia and Marcos Rojo had failed to deliver from the spot, chose to share the glory.
"What's done is done, today is a victory for my team mates because we played a fantastic game and I am very proud to be a member of this team," Tevez told television cameras moments after grabbing the winner.
"Being here means we are doing things very well."
Friday's feeling was very different from that four years ago, when his miss saw Argentina lose to Uruguay in the quarter-finals by the same shootout scoreline. While Tevez sealed the win, Colombian goalkeeper David Ospina was man of the match, making some remarkable stops to keep his side in the game.
The Arsenal goalie made a sensational double save in the 25th minute when he dived to keep out a point-blank shot from Sergio Aguero with his legs and then got up to parry Lionel Messi's header from the rebound.
He also did just enough to tip a shot from Nicolas Otamendi onto the post with 10 minutes left. Jeison Murillo scooped the ball off the line with seconds to go as Argentina piled on the pressure.
Argentina, who have not won the Copa America since 1993, now face the winner of Saturday's Brazil-Paraguay match. The other semi-final will be between hosts Chile and Peru.
(Writing by Andrew Downie; Editing by Peter Rutherford)
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