On a challenging pitch at the Olymp-2 Stadium - which was criticised by United manager Jose Mourinho before the match - midfielder Paul Pogba miscued from inside the box early on.
But United grabbed a vital away goal through Henrikh Mkhitaryan's close-range finish following excellent work by Zlatan Ibrahimovic.
The Swedish striker shot over the crossbar in the second half, before Rostov forward Aleksandr Bukharov latched on to Timofei Kalachev's pass for the equaliser.
Aleksandar Erokhin stabbed a shot wide for the hosts from a promising position and United's Marouane Fellaini headed straight at the goalkeeper from a corner.
The return leg takes place at Old Trafford next Thursday (kick-off 19:45 GMT).
The Brain Behind this Blog
