Jose Mourinho's side, who saw off St Etienne in the previous round, will travel to Russia for the first leg on Thursday, March 9.
The second leg will take place at Old Trafford the following week on March 16.
Mourinho should be relieved to be at home for the return match as United are due to play at Chelsea in their FA Cup quarter-final 72 hours earlier, on Monday 13.
A long trip to Rostov, which is approximately 1,000km south of Moscow, the day after the domestic cup tie would have tested their resolve.
Mourinho has already criticised the scheduling of the Chelsea game on the Monday for TV purposes.
Rostov, who beat Sparta Prague in the last 32, finished second in the Russian Premier League last season.
Elsewhere in the draw, Tottenham's conquerors Gent were drawn against Genk in an all-Belgian tie.
There was also an all-German match-up as Schalke were paired with Borussia Monchengladbach.
Roma, one of the competition favourites, will face Lyon while Celta Vigo were drawn against Krasnodar.
Europa League draw
Celta Vigo v Krasnodar
Apoel v Anderlecht
Schalke v Borussia Monchengladbach
Lyon v Roma
Rostov v Manchester United
Olympiacos v Besiktas
Gent v Genk
FC Copenhagen v Ajax
Europa League odds
Sky Bet make Manchester United the 7/4 favourites to win the Europa League.
Roma are 11/2 second-favourites, while both Schalke and Celta Vigo are priced up at 10/1.
United are odds-on favourites at 8/15 to win the first leg of their tie away at Rostov, who can be backed at 5/1 to win at home.