The Dutch side will be looking to finish the job after winning 3-1 in Serbia last week.
Match preview
Indeed, first-half strikes from Daan Rots, Michel Vlap and Joshua Brenet proved enough to put FC Twente in a commanding position in their Europa Conference League tie.
Stefan Kovac did manage to pull one back after the break to give the Serbs a glimmer of hope heading into the reverse leg, but head coach Ron Jans would have been satisfied with his team's performance and the result they delivered.
Likewise, Sem Steijn's injury-time winner got The Tukkers' domestic season up and running with a victory away to NEC on Sunday, as Jans's side look to improve on last season's impressive fourth-placed finish in the Eredivisie.
First things first, though, they must guard against complacency when Cukaricki arrive into Enschede on Thursday to ensure their European campaign remains on course.
Excluding the Intertoto Cup in the 1990's, this is already the joint-furthest Cukaricki have ever made it in a European competition, making Thursday's second leg something of a free hit for their head coach Dusan Kerkez.
The Belgrade-based side have enjoyed a steady rise within the last decade, registering four third-placed finishes in the Serbian SuperLiga as well as winning the Serbian Cup in 2015.
However, they have not managed to make it to the group stages of either the Europa League or Europa Conference League despite several attempts, and after last week's 3-1 defeat at home to FC Twente, that run looks unlikely to come to an end this season.
Kerkez will be looking to remove any pressure from his players' shoulders ahead of their trip to the Netherlands, with very little expected of them given the contrast in resources between the two sides.
However, an early goal in his side's favour could significantly change the complexion of the tie after Kovac's goal left the door slightly ajar.
FC Twente Europa Conference League form:
FC Twente form (all competitions):
Cukaricki Europa Conference League form:
Cukaricki form (all competitions):
Team News
FC Twente will be without their captain Wout Brama due to a groin injury.
Jans made four changes for the weekend's dramatic triumph away to NEC, so Ricky van Wolfswinkel, Mees Hilgers, Virgil Misidjan and Rots could all be in line for recalls to the starting XI as their boss shuffles the deck.
Cukaricki, meanwhile, will travel without the ineligible duo of Aleksa Jankovic and Milan Savic.
Despite defeat, Kerkez is unlikely to make any changes from last week's side, with no injury concerns restricting his team selection.
FC Twente possible starting lineup:
Unnerstall; Brenet, Hilgers, Pleguezuelo, Smal; Zerrouki, Sadilek; Misidjan, Vlap, Rots; Van Wolfswinkel
Cukaricki possible starting lineup:
Micovic; Stefanovic, Drezgic, Vujadinovic, Roganovic; Docic, Mijailovic; Ivanovic, Kovac, Ndiaye; Badamosi
We say: FC Twente 2-0 Cukaricki
We expect a comfortable FC Twente victory to wrap up their progression to the next stage of the Europa Conference League.
Cukaricki only gained a foothold on home soil once their opponents had taken their foot off the gas having gained a three-goal lead, so we cannot envisage them pushing the Dutch side away from home.