The Armenian gave United the lead after 29 minutes of Sunday's Premier League game to build on his recent string of encouraging performances, but had to be carried off on a stretcher after going over on his left ankle when challenged by Danny Rose in the 85th minute.
But Mourinho has calmed fears Mkhitaryan will be out for a while, predicting he will miss their next two games against Crystal Palace and West Brom but should return for the Boxing Day clash against Sunderland.
He told MUTV: "It looks like the next match no, but it looks like nothing really dangerous or needing surgery.
"Hopefully we will have Mickey on Boxing Day."
Mkhitaryan's goal marked his first in the Premier League since joining from Borussia Dortmund and came after he opened his account for the club in the 2-0 win over Zorya Luhansk on Thursday that sealed United's passage into the last 32 of the Europa League.