The game began slowly, with neither team making much progress in their opponents' half for much of the early stages.
Roma soon began dominating proceedings as Pastore saw a goalbound effort thwarted by a last-ditch block, in addition to Aleksandar Kolarov and Dzeko hitting the woodwork in quick succession.
Neither side could break the deadlock come the half-time mark, while shortly after the break, Torino forward Iago Falque found the back of the net but VAR ruled the goal out for offside.
Both teams enjoyed numerous chances in a lively second half but Roma struck what turned out to be the winner one minute from time.
Substitute Justin Kluivert was the architect, charging to the byline and getting a cross in which fell nicely for Dzeko to emphatically thump home on the volley.
TORINO (3-5-2): Sirigu; Moretti, Nkoulou, Izzo (Bremer 79'); De Silvestri (Aina 25'), Baselli, Meite, Rincon (Soriano 85'), Berenguer; Falque, Belotti
ROMA (4-3-3): Olsen; Florenzi, Fazio, Manolas, Kolarov; Pastore (Schick 75'), De Rossi, Strootman; Under (Kluivert 69'), Dzeko, El Shaarawy (Cristante 59')