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United, still six points behind fourth-placed Manchester City, led when Mkhitaryan raced on to Ander Herrera's through ball and hammered into the roof of the net.
Victor Wanyama should have levelled with a header from six yards out, although visitors Spurs created few chances despite plenty of possession.
Paul Pogba almost made it more comfortable for Jose Mourinho's side when he hit a free-kick against the post, but United ended a run of three games without a win in the league.
The game ended in disappointment for Mkhitaryan, who was taken off on a stretcher with an ankle injury in the final 10 minutes at Old Trafford and has been ruled out until Boxing Day.
Spurs missed the chance to replace Manchester City in fourth and remain fifth, three points above United.
2016.12.12 / 09:12