United, considered the favorite by many bookmakers, is the only British side left in Europe's secondary club competition, BBC reported.
Jose Mourinho's side beat French side Saint-Etienne 4-0 on aggregate in the last 32.
"It is a bad draw in every aspect," said Mourinho. "It is far and difficult — and comes in a bad period."
The two-legged ties will be played on March 9 and March16.
A busy March for United:
Mar 4 Bournemouth (H) Premier League
Mar 9 FC Rostov (A) Europa League round of 16
Mar 13 Chelsea (A) FA Cup quarter-finals
Mar 16 FC Rostov (H) Europa League round of 16
Mar 19 Middlesbrough (A) Premier League
Last season's Russian Premier League runner-up Rostov is the lowest-ranked team left in the Europa League, with UEFA'S system rating them as only the 93rd best team on the continent.
It is competing in the last 16 of a major European competition for the first time.
Rostov is seventh in its domestic league at the midseason winter break, having won two of its previous eight league matches.
A 4-0 first-leg victory against last-32 opponent Sparta Prague set up a comfortable 5-1 aggregate win and booked its first meeting against three-time European champion United.
It dropped into the Europa League after finishing third in a Champions League group containing Atletico Madrid, Bayern Munich and PSV Eindhoven.
"They had a very good Champions League campaign," added Mourinho.
"They beat Ajax and Anderlecht in qualifiers, managed important results against Bayern and Atletico, got third position to knock PSV out.
"The team is very defensive and physical. A bad draw."
The full draw:
Celta Vigo (Spa) vs. FC Krasnodar (Rus)
Apoel Nicosia (Cyp) vs. Anderlecht (Bel)
Schalke (Ger) vs. Borussia Monchengladbach (Ger)
Lyon (Fra) vs. Roma (Ita)
FC Rostov (Rus) vs. Manchester United (Eng)
Olympiakos (Gre) vs. Besiktas (Tur)
Gent (Bel) vs. Genk (Bel)
FC Copenhagen (Den) vs. Ajax (Net)