Argentina dominated throughout but found Arsenal keeper David Ospina in fine form as he denied Sergio Aguero, Lionel Messi and Nicolas Otamendi.
That meant penalties were needed and, after Colombia's Jeison Murillo missed, Tevez converted to seal a 5-4 victory.
Argentina face either Brazil or Paraguay in Tuesday's semi-final.
It capped a memorable day for Tevez, who came off the bench to score moments after a move from Juventus to his hometown club Boca Juniors was confirmed.
It was also a deserved win for Argentina who, having scored just four goals in their three previous games at the tournament, displayed their attacking intent from the outset.
They embarked on wave after wave of attacks but found themselves up against a Colombia side that had set up to frustrate their opponents.
And when Argentina did find their way through the Colombia defence, Ospina was there to deny them.
The goalkeeper, whose future at Arsenal is in doubt with the imminent arrival of Petr Cech from Chelsea, produced a stunning double save to keep out Aguero's shot from close range and then somehow get back up to fling himself at Lionel Messi's header on the follow-up.
That gave Colombia confidence, and they finally managed to muster a shot on target in the second half when Jackson Martinez powered a header towards goal, but Sergio Romero got down to make the save.
However, it was left to Ospina to keep Colombia in the game as he made another crucial save, this time denying Otamendi when the defender volleyed a corner at goal and Ospina pushed the ball onto his post.
With no extra-time in the Copa America quarter-finals or semi-finals, the game went to penalties and, after five were missed, Tevez stepped up to seal victory.