Juan Mata got the decisive goal for United when he stabbed in from Zlatan Ibrahimovic's flick, but Rostov threatened to take the game to extra time and Sergio Romero made two good saves late on.
The United goalkeeper first kept out Sardar Azmoun's flicked header before thumping away Christian Noboa's free-kick.
It was a largely frustrating night for United, who dominated for large periods without really threatening and also lost midfielder Paul Pogba to a hamstring injury.
Their best chances before the goal came in the first half, when Henrikh Mkhitaryan shot wide when one-on-one and Ibrahimovic twice hit the post.
United will find out on Friday who they will play in the quarter-finals, with the draw taking place at 12:00 GMT.