The Rossoneri were tasked with responding after Inter's 2-0 win over Hellas Verona on Sunday (AEST), which moved the Nerazzurri just a point adrift.
But, after seeing fellow challenger Napoli slip up with a 3-2 home defeat to Fiorentina, Milan also stumbled.
Clear-cut chances were few and far between, Milan failing to take advantage of its edge in possession.
It means Inter can go a point ahead at the top with a win at Spezia on Saturday, when it kicks off two hours before Milan hosts Genoa.
Davide Calabria's long-range effort drew a strong save from Etrit Berisha in the 24th minute before Torino midfielder Samuele Ricci fired wide of the near post at the other end.
Mike Maignan tipped over from Mergim Vojvoda five minutes after the restart as Torino created the better opportunities in the second half.
Sandro Tonali was denied by Berisha after cleverly working his way into the box, with Andrea Belotti seeing a chance to secure a shock win for Torino go begging when he fired into the side-netting on an evening where the Granata were full value for a stalemate that could prove pivotal in the battle for the Scudetto.