Tevez, who missed the crucial penalty when Argentina went out in a shoot-out at the 2011 tournament, rammed homed the winning spot-kick after Colombia's Jeison Murillo had blazed over the bar.
The 31-year-old, who came on as a second half substitute, provided the dramatic final twist to a shoot-out which saw Lucas Biglia and Marcos Rojo both miss kicks for Argentina that would have sealed victory after regulation time finished locked at 0-0.
Argentina's win puts them into a semi-final against the winner of Saturday's quarter-final between Brazil and Paraguay.
"In soccer there's always a second chance," said Tevez, who earlier Friday sealed his move back boyhood club Boca Juniors in Buenos Aires.
"But tonight is not about me, it's about my team-mates. My team-mates won this match and I'm very proud of them, and I'm proud to be part of this squad."
Tevez, who spent three long years frozen out of Argentina's national team before returning after the 2014 World Cup, said he was happy with his non-starting role in the Albiceleste.
Argentina coach Gerardo Martino admitted that he had deliberately omitted Tevez from his list of five designated penalty takers for the shoot-out in order to ease pressure on the former Manchester City striker.
"I didn't put Tevez among the top five because the last time he shot, Argentina went out. We wanted to avoid that," Martino revealed.
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