Mkhitaryan scored the winning goal against Tottenham on Sunday and is finally displaying the sort of form that had convinced United to sign him from Borussia Dortmund in the close-season.
But late in the game at Old Trafford, the Armenian went down under a foul from Spurs defender Danny Rose and had to be carried from the field with what appeared a nasty injury to his ankle.
Speaking on Monday about the incident, Mourinho said he had feared a lengthy absence was likely as it looked a serious injury.
But to the relief of the United manager, Mkhitaryan should miss just two games, away at Crystal Palace and West Bromwich Albion this week, before returning for the busy holiday schedule.
"Hopefully he can play at Christmas. We believe it's possible," Mourinho told reporters.
"When I saw him on the stretcher I thought it would be more difficult. When I watched the tackle on TV, I was a bit scared.
"It's a pity because he is going to miss matches in his best period but at least we don't lose a player for so long."
The Spurs game was just Mkhitaryan's third league start for United and his first goal in the English top flight after the former Bundesliga player of the year initially struggled to adapt to his new surroundings.
A disastrous debut did not help in the home derby defeat to Manchester City in September, but Mourinho is now convinced he is about to reap the benefits of allowing him to take his time adapting to the English game.
"I have had players with a similar adaptation process. But we bought him, he was not a player chosen by other people," Mourinho said.
"We bought him because I asked the club to do it because I liked the player. I analysed the player so it would be more frustrating for me if everything went wrong.
"But he needed the period of adaptation at every level and in that period he was a great professional because maybe another guy in this position would find excuses.
"Another player would blame the manager, he would blame the pitch, he would blame the weather, he would blame something, like many of the players now do.
"Mkhitaryan was quiet, he closed his mouth, worked really hard waiting for this opportunity and is playing so well now.
"But hopefully it's not a few matches out that are going to stop his evolution."