The Armenia international finished off Zlatan Ibrahimovic's cutback to open the scoring for the Premier League side on his return from injury, having not featured since facing Saint-Etienne in the last round.
But Aleksandr Bukharov equalised for Rostov early in the second half to leave the tie in the balance, with Mkhitaryan highlighting the surface as a problem for United.
"We tried to win today, maybe we were not focused at the beginning of the second half, but we played as good as we could on this pitch," Mkhitaryan told BT Sport.
"I don't want to even speak about the pitch, because it was an awful thing for both sides. It was not a very interesting game but whatever, we tried to do everything."
6 - Henrikh Mkhitaryan has been directly involved in six goals in his last seven appearances for Man Utd (4 goals, 2 assists). Return.— OptaJoe (@OptaJoe) March 9, 2017
Rostov had the better chances to win the game, with Mkhitaryan acknowledging Bukharov's effort changed the dynamics of the game.
"You know that's football, you have to wait for everything, it doesn't matter if you score and you lead 1-0, you have to be ready for everything," the playmaker said.
"We conceded the goal and that was a mistake but there is a second leg to play which will be more important.
"The first half was more quiet and there was nothing special besides the goal. I'm happy to score and we were happy to lead, but unfortunately we conceded and everything changed with their goal.
"Hopefully I've got no injuries so we'll keep fighting and getting ready for the next game."