Marcelo Larrondo bagged a double before Paulo Vitor Barreto put the seal on a commanding win for the Granate in their first European outing for two decades
Having waited 20 years for their return to European football, the Serie A side assumed total control of their third qualifying round tie with three away goals.
Marcelo Larrondo opened the scoring just before half-time with a well-taken penalty after Pontus Segerstrom was dismissed for upending Giuseppe Vives.
The Argentine striker added a second eight minutes after the break and Paulo Vitor Barreto put them in total control in the 58th minute.
Brommapojkarna were handed a way back into the match when Vives was shown red for a foul on Serge Martinsson-Ngouali inside the penalty area, but Dardan Rexhepi missed the subsequent spot-kick to leave Torino in total control of the tie.