Bravo, who missed the first two games because of injury, stopped penalties by Ricardo Quaresma, Joao Moutinho and Nani, sending Chile to its third straight international final after winning consecutive Copa America titles.
Arturo Vidal, Charles Aranguiz and Alexis Sanchez all scored for Chile in the shootout.
The teams had drawn 0-0 after normal and extra time.
The result was justice for Chile, which had a strong penalty appeal turned down in extra time and then saw the ball hit the woodwork twice in the same move a few minutes later.
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Chile will play Germany or Mexico in Sunday's final in St. Petersburg. They will be seeking their first title outside of South America after winning the Copa America in 2015 and 2016.
Portugal was looking for their second consecutive title after winning the European Championship last year.
In the shoot-out, Bravo dived to his right to stop weak penalty kicks by both Quaresma and Joao Moutinho, then switched sides to save Nani's timorous shot. Nani tip-toed desperately slowly to the ball looking for all the world as though he did not want to take the kick. And he patted it gently to Bravo.
Chile had their chances to win it in extra time when Vidal's powerful right-footed shot struck the far post. Off the rebound, substitute Martin Rodriguez hit the crossbar with Portugal goalkeeper Rui Patricio already beaten.
Sanchez had already had a chance to break the deadlock six minutes into extra time, but his header from near the penalty spot missed just wide after a well-placed right-flank cross by Mauricio Isla.
Chile wanted a late penalty when Francisco Silva was stepped on by a defender inside the area with only a few minutes left, but the referee let the play continue and it was not reviewed by video.
It was the first time in a major international tournament that extra time was played and teams were allowed to make a fourth substitution. Portugal coach Fernando Santos made the fourth change in the final minutes, replacing Andre Gomes by Gelson Martins. Chile only made three substitutions.
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