The Premier League side had to fight hard to establish a first-leg advantage in a thrilling play-off tie.
But a Bremer own goal and second-half strikes from Diogo Jota and Raul Jimenez rewarded Wolves' positive approach, with Lorenzo De Silvestri and Andrea Belotti providing hope for Torino at Molineux next Thursday.
Wolves kept seven of the side which had started the 1-1 Premier League draw against Manchester United on Monday.
Matt Doherty, Ryan Bennett, Jonny and Ruben Neves were the four players to make way for Jesus Vallejo, Romain Saiss, Adama Traore and Ruben Vinagre.
Wolves had beaten Crusaders of Northern Ireland and Armenian outfit Pyunik to reach this stage, but this was by far the sternest European test so far for Nuno Espirito Santo's side.
The opening stages were played at a frantic pace with Torino determined to get on the front foot.
Simone Zaza was off target with an early header, but Wolves almost went ahead when Traore burst down the left in the 13th minute.
Traore cut the ball back to Jota near the penalty spot, but his goalbound effort was hacked away by Torino goalkeeper Salvatore Sirigu.
Chances continued to come at both ends as Nicolas Nkoulou shook the Wolves crossbar before Leander Dendoncker headed wide from six yards.
Wolves escaped again when Armando Izzo was left unmarked at the far post and Rui Patricio saved well with his legs.
It was to prove a vital intervention as the visitors scored a precious away goal three minutes later.
Joao Moutinho flighted his free-kick towards a congested penalty area and. although Saiss was in close proximity, it looked as if Torino's Brazilian defender Bremer had applied the final touch.
Torino raised the intensity at the start of the second half, but Wolves were compact and organised with a black and gold wall blocking the path of the Italians.
Traore was also eager to show his frightening pace and power, and Cristian Ansaldi and Alex Berenguer were both cautioned for crude fouls on the Spaniard.
Wolves doubled their lead after 59 minutes when Traore worked his way to the by-line and pulled the ball back for Jota to sidefoot home from 10 yards.
Torino kept the tie alive when the Wolves defence switched off two minutes later, allowing the unmarked De Silvestri to meet Ansaldi's deep cross two minutes later with a firm header.
But Jimenez restored Wolves' two-goal lead after 72 minutes when the Portuguese referee Artur Dias allowed play to continue after the substitute Pedro Neto was clearly fouled.
Jimenez strode on past some tame Torino tackling to find the bottom corner of Sirigu's net for his fifth goal in the competition.
Belotti gave Torino hope for the second leg by converting an 89th-minute penalty after he had been fouled by Vinagre.
The Italy forward sent his spot-kick down the middle past Patricio and he almost equalised seconds later.
But the Portuguese goalkeeper pushed Belotti's low fizzing strike past a post for Wolves to secure the win that their performance had deserved.