Manchester United took the lead when Mkhitaryan, who finally has the trust of Jose Mourinho, was released by Ander Herrera and finished cleanly.
In the second half, Spurs threatened as Matteo Darmian was targeted by Moussa Sissoko, but United held on.
Another blow for United came late on when Mkhitaryan left on a stretcher following a hack from Danny Rose.