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.
"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 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.
They 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."
Manchester United have been drawn against Russian side FC Rostov in the Europa League last 16.