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 Argentinian striker added a second eight minutes after the break and 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.
A Kehinde Fatai hat-trick saw Romanian side Astra beat Slovan Liberec 3-0, while Omonia Nicosia matched that scoreline at home to Metalurg in Cyprus.
Three-goal winning margins were popular on Thursday as Elsfborg and Lyon both recorded 4-1 victories of FH and Mlada Boleslav, while Rijeka and Krasnodar went one better as they beat Vikingur and Diosgyor 5-1.
PSV, winners of the competition in its former guise of the UEFA Cup in 1978, started their campaign with a solid 1-0 victory over St Polten thanks to Luuk de Jong, while two-time winners IKF Gothenburg lost by the same score to Primeira Liga side Rio Ave.
Dinamo Minsk, Young Boys, Mainz and Stjarnan all won their respective matches 1-0, while Shakhter Karagandy beat Hajduk Split 4-2 in Kazakhstan.
Late goals rescued 1-1 draws for Molde and Viktoria Plzen, but Dinamo Moscow and AIK will be disappointed to be held by Ironi Kiryat Shmona and Astana.
Real Sociedad and Split gave themselves a good cushion for their return matches next week with 2-0 triumphs, while Spartak Trnava and Zimbru conceded late goals against Grodig and St Johnstone as they won 2-1 away from home.
Shakhtyor can begin preparing for a play-off place after producing a stunning performance in Belgium to beat 10-man Zulte-Waregem 5-2, but Brondby were not so lucky as goals from Oscar Duarte, Nicolas Castillo and Victor Vazquez gave Club Brugge a 3-0 triumph.
Meanwhile, Evangelos Nastos struck Atromitos' second goal of the game in Bosnia-Herzegovina as they beat Sarajevo 2-1
Hull City's first foray into Europe ended in a goalless draw against Trencin, although things could have been much different had Tom Huddlestone not missed a second-half penalty.
The Premier League side battled out one of four goalless draws around Europe as Rosenborg, Esbjerg and Chikhura held Karabukspor, Ruch Chorzow and Neftci respectively.