The visitors went ahead when Joao Moutinho’s free-kick was headed into his own net by Torino’s Gleison Bremer.
Diogo Jota doubled the advantage as he finished after Adama Traore’s pass.
Lorenzo de Silvestri pulled one back, Raul Jimenez’s calm finish restored Wolves’ two-goal lead before Andrea Belotti’s late penalty for Torino.
The two teams meet again at Molineux on 29 August with the winners of this play-off tie moving into the main group phase of the competition.
Wolves, playing in their first European campaign since 1980-81, will advance if they avoid defeat or only lose by one goal in a low-scoring second leg.
Wolves win despite tough schedule
Nuno Espirito Santo will be delighted with the performance of his team, who were playing for the second time in four days after drawing 1-1 against Manchester United in the Premier League, while Torino are yet to start their Serie A campaign.
This was the English side’s seventh match in 29 days and Nuno opted to rest midfielder Ruben Neves, who had scored a wonderful goal against United.
But the gamble paid off as Wolves edged a thrilling game at the Stadio Olimpico Grande in Turin.
Indeed, Nuno may even be disappointed they do not have a two-goal advantage as Belotti’s 89th-minute penalty – following a clumsy challenge on him from Ruben Vinagre – gave the Italians a lifeline.
In added time, Torino had a chance to snatch an equaliser but goalkeeper Rui Patricio produced a fine save to deny Belotti his second goal.
Lightning-quick Traore impresses again
The half-time introduction of the rapid Traore changed the game against United on Monday and the Spaniard was rewarded for his efforts with a place in the starting line-up in Italy.
Operating again at right-wing back, his speed and directness caused Torino serious problems, with Wolves nearly scoring after 12 minutes when Traore’s run and cross found Jota, whose low shot was kept out by the legs of home goalkeeper Salvatore Sirigu.
Nicolas Nkoulou headed against the bar for the hosts before Sirigu was beaten, albeit by one of his own players as Bremer headed in following Moutinho’s excellent set-piece delivery.
Traore was heavily involved again as Wolves doubled their lead after another direct run and a well-timed Cruyff turn gave him space to set up Jota who this time converted the chance.
However, two minutes later Torino grabbed their first goal with De Silvestri heading in from Cristian Ansaldi’s cross.
But it was not long before Wolves were two goals in front again, with the referee playing a good advantage after Pedro Neto was fouled and Jimenez running through the defence to beat Sirigu for his fifth goal in this Europa League campaign.
Wolves had reached this phase with a 6-1 aggregate win over Northern Ireland’s Crusaders and then an 8-0 overall success against Pyunik of Armenia.
Torino, who finished seventh in Serie A last season and are managed by former Watford boss Walter Mazzarri, were always going to be a tougher challenge and Belotti’s penalty gives the Italians hope for the second leg.
Man of the match – Adama Traore (Wolves)
‘Far from being over’ – what they said
Wolves manager Nuno Espirito Santo said: “It’s far from being over, it’s a close tie.
“We played good, we were organised, we stayed in shape, we produced some goals but, like I say, there’s still one game to go and we have to be really, really good to perform again against a very, very good team.”
On the performance of Traore, Nuno added: “I’m pleased. He helped the team, others did what they had to do. It’s about tasks, it’s about rules, it’s about always being a team.”
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