He has signed a two-year contract to replace Juric, who announced in May that he would leave the club after his contract expired at the end of the season.
Vanoli led Venezia back to Serie A this season after a two-year absence, securing promotion from Serie B via the playoffs.
Torino finished the Serie A campaign in ninth place, narrowly missing out on a European place.
While Venezia have not confirmed a replacement, Italian media reports said former Frosinone coach Eusebio Di Francesco is set to take over as their manager.