River, who lost the first leg 1-0 at home, went a goal down against the run of play after 35 minutes when Leonardo put the Brazilians ahead.
The Buenos Aires side kept pushing forward, however, and Rafael Borre scored the equaliser with 81 minutes gone before the referee awarded them a penalty five minutes later after consulting with the video assistant referee (VAR).
Gonzalo Martinez converted after a long delay and a red card for Gremio's Bressan and although the referee added 14 minutes of stoppage time the home side could not get another goal.
"We believed all the way until the final minutes, with faith and will to win," said River's Javier Pinola. "We deserved to win."
The remarkable victory sets up the possibility of an all-Argentine final against Boca Juniors, who play Palmeiras in Sao Paulo on Wednesday having won the home leg 2-0.
(Reporting by Andrew Downie; Editing by Peter Rutherford )
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