The Italian outfit reached the final of this competition in its previous guise of the UEFA Cup in 1992, before going down on away goals to Ajax.
They played in the Intertoto Cup third round back in 2002 but last featured in a top-level European competition when being edged out in a UEFA Cup Winners' Cup quarter-final against Arsenal in 1994.
Tony Adams' goal proved the only difference that day as Arsenal progressed 1-0 on aggregate, and Torino have failed to grace the European stage ever since.
It has been a tumultuous period for the Turin club in that time - plagued by financial troubles, they have bounced between the top two tiers in Italy and remain in the shadows of their illustrious neighbours Juventus.
Thursday's first leg at home represents a chance for Giampiero Ventrua's side to distinguish themselves once more, and defender Cesare Bovo will not be taking the Swedish outfit lightly.
"We have been working for a month just to overcome this hurdle, we want to go through at any cost," he told UEFA.com.
"We know that our opponents play with a high tempo and intensity, but we want to do well, we are self-confident and we won't underestimate them."
Ventura has confirmed that he is likely to be without Alessio Cerci and Matteo Darmian for the clash, with both having only just returned to training after their World Cup exploits with Italy.
Meanwhile, the third qualifying round also sees other big names enter the competition, with Dutch giants and 1978 winners PSV hosting St Polten of Austria, while 2010 UEFA Champions League semi-finalists Lyon will visit Mlada Boleslav of the Czech Republic.
Real Sociedad - recently shorn of star striker Antoine Griezmann, who has joined Atletico Madrid - face Scottish side Aberdeen and two-time champions Gothenburg are also still involved, with the Swedish outfit taking on Portugal's Rio Ave.
Rosenborg, former Champions League quarter-finalists, travel to Turkey to face Karabukspor, and Brondby visit Club Brugge.
Steve Bruce's Hull City, FA Cup runners-up in May, begin their European adventure in Slovakia against Trencin, with the Premier League side sweating on the fitness of Alex Bruce (Achilles), Curtis Davies (hamstring) and David Meyler (knee).
"We hope that they are going to be involved," said the Hull boss. "They are all a concern at this moment as it is only a few days away and it is a European tie."
Hull will be wary of an upset on their travels, and there are similarly awkward away legs for AIK, Slovan Liberec, Esbjerg and Viktoria Plzen - who face Astana of Kazakhstan, Astra of Romania, Ruch Chorzow of Poland and another Romanian side, Petrolul Ploiesti, respectively.
Mainz, who finished seventh in the Bundesliga last season, play host to Asteras Tripolis from Greece and Scottish side St Johnstone, who needed penalties to see off Luzern in round two, take on Spartak Trnava.
Home advantage could well prove crucial when Split host Chornomorets, Diosgyor welcome Krasnodar, Omonia Nicosia take on Metalurg, Stjarnan of Iceland face Lech Poznan and Zorya of Ukraine prepare to meet Molde.
Sarajevo are in action against Greek side Atromitos while Dinamo Moscow face Hapoel Kiryat Shmona, Young Boys welcome Ermis to Switzerland and Hafnarfjordur travel to Elfsborg.
Zulte Waregem of Belgium made it to the group stages last year, and they will aim to continue their progress when they face Shakhtyor of Belarus.
Elsewhere, Grodig of Austria take on Zimbru after a 5-2 aggregate success over Cukaricki in round two, while Faroese side Vikingur - unbeaten in their last six games in all competitions - face a tough test against Rijeka.
Shakhter Karagandy of Kazakhstan finished bottom of Group L in last year's competition, and their bid to go better in 2014-15 continues at home to Hadjuk Split.
Dinamo Minsk take on CFR Cluj in their first-leg clash, with Neftci at home to Chikhura of Georgia.
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