However, the first-half stoppage time proved to be the visitors' undoing as Torino's Argentine defender Cristian Ansaldi netted the equalizer in the second minute of extra time.
With its fifth win of the season, Torino provisionally climbed up to the sixth spot in the Serie A standings with 21 points, while Genoa remained in 14th place with 15 points.
In other Italian league action, eighth-placed Sassuolo drew 0-0 at home against 17th-placed Udinese.
The 19th-placed Frosinone played to a 1-1 tie with 13th-placed Cagliari.
Frosinone midfielder Francesco Cassata scored in the 14th minute off an assist from Francesco Zampano, but Cagliari forward Diego Farias equalized in the 77th minute with help from attacking midfielder João Pedro.
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