Former Paris Saint-Germain star Zlatan Ibrahimovic struck the second in the closing stages to send United through in second place in Group A behind Fenerbahce, who won 1-0 at Feyenoord thanks to a Moussa Sow effort.
United's game went ahead despite concerns about the state of the pitch at the Chornomorets Stadium in Odessa.
Jose Mourinho made six changes to the side that drew 1-1 at Everton on Sunday in the English Premier League, including a return for skipper Wayne Rooney.
"I was waiting a long time for this goal," said Mkhitaryan who moved to Old Trafford in August amid great fanfare for 35 million euros from Borussia Dortmund. "My next goal has to be at Old Trafford because I want to score at home," the Armenia international told BT Sport.
"The first half was a little bit difficult because we couldn't find the last pass but in the second half we created some space for the wingers and the strikers. The important thing was to win today and go through."
United, who are sixth in their championship after 14 games, are 13 points behind pacesetters Chelsea and entertain Tottenham Hotspur on Sunday.