The Granata took a while to get into gear against Palermo, who were just promoted to Serie B and are currently under a caretaker manager after Silvio Baldini walked away.
The Sicilians started strong, with saves on Roberto Floriano and Matteo Luigi Brunori, but Torino broke through instead when Nemanja Radonjic went on a mazy run, setting up Sasa Lukic from the edge of the box.
Radonjic had a goal ruled offside before doubling their lead with his header on a Lukic cross.
Pietro Pellegri sealed the scoreline from close range in a chance created by Wilfried Singo.
Toro will therefore face Lecce in the Second Round, to be played in October.
Empoli have already been eliminated from the Coppa Italia at the first hurdle by Serie B club SPAL, who snatched it in the final seconds of extra time.
Now SPAL will play either Genoa or Benevento in the next round.
Empoli 1-2 SPAL (aet)
Cambiaghi 80 (E), Dickmann 82 (S), Arena 121 (S)
Torino 3-0 Palermo
Lukic 54 (T), Radonjic 73 (T), Pellegri 79 (T)