The Premier League side however will want to go through in style with victory, even if Europe's second-tier competition does not feature at the top of their priorities in Mourinho's first season in charge.
The Old Trafford side will also be eager to end a bout of frustrating draws, the latest being a 1-1 stalemate at Everton at the weekend when Mourinho's men conceded an 89th-minute equaliser from the spot.
The Zorya match -- Thursday is the last phase of Europa League group games -- will take place in Odessa, southern Ukraine, a four-hour flight from northern England and where temperatures will hover around freezing by evening kickoff.
Zorya might already be out of the competition, but United's French forward Anthony Martial is not expecting the Ukrainians just to lie down.
"It is very different, not just the supporters but the weather. Over there, it's really cold at this time of the season," Martial told MUTV.
"But it's down to us to come to terms with that, adapt to the environment we find ourselves in and get the victory we need."
He added: "We're expecting a difficult encounter. Things went quite well for us in the home game (1-0 to United) but it'll be different over there.
"They'll be in front of their own supporters and will be determined to make life difficult for us to get a result, so we'll be on our guard."
United will also go through to the last 32 with defeat if Feyenoord fail to beat Fenerbahce at home in the other Group A fixture.
Fellow Premier League representatives Southampton face the Israeli side Hapoel Beer-Sheva at St Mary's Stadium knowing that victory or a 0-0 draw will put them in the last 32, where eight clubs who failed to progress from the Champions League will enter the fray.
"It's a huge game on Thursday and that's all anyone is worried about now," young midfielder Sam McQueen, fresh from signing a new four-and-a-half-year contract with the club, told Southampton's website.
"It's a huge competition. You're playing against a lot of different styles, which is only going to help me grow personally, as well as the team against new challenges.
"I think we've done well so far, but we wouldn't be doing ourselves justice if we don't win and get through the group."
They may not be European household names but Ireland's Dundalk and Qarabag of Azerbaijan can make a slice of history.
No Irish side has ever reached the knockout stages of a UEFA club competition but domestic champions Dundalk can change that if they win at Maccabi Tel Aviv and AZ Alkmaar do not at home to Zenit St Petersburg.
A score draw for the Irish would also suffice if AZ lose in the Netherlands.
Qarabag will definitely qualify with a home victory in Baku over Fiorentina, who are all but through.
The European minnows, who are in the Europa League group stage for a third straight campaign but have yet to make it to the round of 32, are impressive at home.
Qarabag are unbeaten in five European home matches since a 1-0 loss to Tottenham Hotspur in last season's group stage.
In all, 11 last-32 places are still up for grabs.
Ajax, Roma, Zenit and Schalke are among the sides already through, but Inter Milan are early casualties with nothing to play for.