The Red Devils faced a difficult 3,750-mile round trip for the game in south Russia – and they were cheered on by 238 travelling supporters, who each had their visas paid for by the club and were given blankets on entering the ground.
Mourinho set up with three centre-backs and Ashley Young and Daley Blind acting as wing-backs in a change of formation for the Premier League side.
The Portuguese manager spoke before the game about deploying a more “direct” approach because of the dreadful pitch, but he may also have had Monday’s FA Cup quarter-final against his old club Chelsea in mind.
As well as a dry and bobbly surface, the stop-start game – which had a total of 38 fouls – made for a poor spectacle. However, Mourinho will surely be confident his players can go back to Old Trafford and complete the job next week.
For their goal, Fellaini held off a home defender before feeding Ibrahimovic, whose quick feet allowed him to poke the ball into team-mate Mkhitaryan’s path and the Armenia international struck for the third time in this season’s competition.
Rostov entered the Europa League after finishing third in their Champions League group behind Atletico Madrid and Bayern Munich. In fact, they managed to beat the German champions at home – though that was their only victory from six games.
A team without any household European names, they struggled to impose themselves against far superior opponents in the opening period.
But on 53 minutes, Kalachev’s raking pass sailed between defenders Phil Jones and Chris Smalling and Bukharov calmly controlled the ball on his chest and slotted in.
Skipper Aleksandr Gatskan struck a long-range shot straight at Sergio Romero late on, but the draw meant Rostov have lost just one of their past eight games at home in Europe.
Man of the match – Marcos Rojo (Manchester United)
A throwback to the amateur days
Manchester United manager Jose Mourinho, speaking to BT Sport: “It was a very good performance in relation to the conditions. It was impossible to play better, impossible to play a passing game.
“We played what the game demanded and we played well. We made one defensive mistake.
“l remember as a kid some matches like this in Portugal – non-league and amateur pitches. To see my players coping with it and the humility to fight for very ball is a good feeling for me.
“We have an open result for the second leg with a little advantage for us. There are no injuries.”
Manchester United goalscorer Henrikh Mkhitaryan, speaking to BT Sport: “It does not matter if you are leading 1-0, you have to be ready for everything.
“We conceded the goal and there was a mistake – but we have a second game to come.”
Rostov are back in league action when they take on Terek Grozny on Sunday, while Manchester United visit Chelsea in the FA Cup quarter-final on Monday (kick-off 19:45 GMT).
Talisman Mkhitaryan – the stats
Manchester United have drawn four of their five away trips against Russian opponents in Europe.
Jose Mourinho’s sides have scored an away goal in the first leg of a European tie in 11 of the 13 games he has overseen.
Aleksandr Bukharov’s 53rd-minute strike was the first goal United have conceded in 443 minutes in the Europa League.
The Red Devils recorded their lowest pass accuracy of the season in this match (61.17%).
Henrikh Mkhitaryan is the first United player to score in three successive European games since Wayne Rooney in March 2010.
The Armenian has been directly involved in six goals in his past seven appearances for the club (four goals, two assists).
Zlatan Ibrahimovic provided the assist for the first goal – he has been directly involved in 40% of Manchester United’s 82 goals this season.