Roy Hodgson's Liverpool side will meet the club based in the city of Trabzon, who have won the Turkish league six times, on August 19 and 26.
The play-off is the final hurdle before qualifying for the group stage, with Friday's draw involving 74 clubs.
Villa face Rapid Vienna - Austria's most successful club with 32 championships.
The rivals met in the same round last season with Villa crashing out of the competition on away goals.
Big-spending Manchester City travel to Romania for the first leg of their play-off after being paired with Timisoara.
The Romanian minnows, established since 2002, finished fifth in their domestic league last season.
Celtic will take on Dutch side Utrecht. The Hoops' dropped down to European club football's second tier competition after being knocked out of the Champions League qualifiers by Braga on Wednesday night on a 4-2 aggregate.
Neil Lennon's side, who were seeded for the draw in Nyon, will play the first game at home before travelling to Holland a week later.
Motherwell will fancy their chances of making the group stages for the first time after being drawn to play Odense, with the first leg in Denmark.
Craig Brown's men have come through two qualifying rounds against Icelandic side Breidablik and Aalesund.
The Norwegian side were beaten 3-0 at Fir Park on Wednesday as the home side cruised through to the play-offs with a a 4-1 aggregate win.
Motherwell chairman John Boyle played down his side's chances of getting past Odense, saying: "We are happy just to be in the draw and we will hopefully do our best.
"They will be a formidable side, they all are at this level and you could say we are the minnows.
"It is a decent distance in terms of travel and overall we are pleased but it is still a huge challenge for us."
Dundee United will make their first appearance in the Europa League this season when they take on AEK Athens.
The Scottish Cup holders were drawn at home in the first leg and will be looking to take a decent lead to Greece for the return game.
Welsh outfit TNS also travel for their first leg, heading to Bulgaria to meet CSKA Sofia.
Italian giants Juventus also have Austrian opposition as they take on Sturm Graz while Palermo will begin at home to Maribor and Napoli face Elfsborg.
Villarreal have been drawn against Dnepr-Transmash, although their participation is still subject to an appeal to the Court of Arbitration for Sport from Mallorca, who were disqualified under UEFA's financial regulations.
Spain's other representatives at this stage, Getafe, have been paired with Cypriot side APOEL Nicosia.
Germany have three clubs in the competition at this stage, with Stuttgart paired with Slovan Bratislava, Bayer Leverkusen facing Tavriya Symferopol and Borussia Dortmund drawing the short straw as they face a long trip to take on Azerbaijani side FK Qarabag.
Paris Saint-Germain will face a testing trip to Israel to play Maccabi Tel Aviv and Lille start away to FC Vaslui, while some of the Dutch contingent also got a challenging draw.
Feyenoord take on Gent and FC Utrecht face Scottish giants Celtic, but PSV Eindhoven will feel more confident about their draw against Sibir while AZ Alkmaar got Aktobe.
Source: Team Talk
Source: Team Talk