Henrikh Mkhitaryan gave the Premier League team a 35th minute lead, turning the ball in from close range after some nifty footwork by Zlatan Ibrahimovic.
But Rostov, who beat Bayern Munich on the same ground in November on their way to a third-place finish in their Champions League group, levelled eight minutes after the restart.
Alexander Bukharov volleyed past Sergio Romero for a superbly-taken goal after chesting down a long ball over the United defence from Timofei Kalachev.
Both sides struggled on a surface which had been heavily criticised by United manager Jose Mourinho and dust was getting into the eyes of the players towards the end of the game.
In other early kickoffs, FC Copenhagen's Andreas Cornelius headed in on the hour to secure a 2-1 home win over former European champions Ajax Amsterdam, while Anderlecht won 1-0 at APOEL Nicosia thanks to a first-half goal from Nicolae Stanciu.
The other first-leg ties were being played later on Thursday, including AS Roma's trip to Olympique Lyonnais and Schalke 04's home clash with Borussia Moenchengladbach.
