United, considered the favourites by many bookmakers, are the only British side left in Europe's secondary club competition.Jose Mourinho's side beat French side Saint-Etienne 4-0 on aggregate in the last 32.
The two-legged ties will be played on Thursday, 9 March and Thursday, 16 March.
Last season's Russian Premier League runners-up Rostov are the lowest-ranked team left in the Europa League, with Uefa's system rating them as only the 93rd best team on the continent.
They are competing in the last 16 of a major European competition for the first time.
Rostov are seventh in their domestic league at the mid-season winter break, having won two of their previous eight league matches.
A 4-0 first-leg victory against last-32 opponents Sparta Prague set up a comfortable 5-1 aggregate win and booked their first meeting against three-time European champions United.
The full draw
Celta Vigo (Spa) v FC Krasnodar (Rus)
Apoel Nicosia (Cyp) v Anderlecht (Bel)
Schalke (Ger) v Borussia Monchengladbach (Ger)
Lyon (Fra) v Roma (Ita)
FC Rostov (Rus) v Manchester United (Eng)
Olympiakos (Gre) v Besiktas (Tur)
Gent (Bel) v Genk (Bel)
FC Copenhagen (Den) v Ajax (Net)