Mkhitaryan, 27, settled the game with his first Premier League goal, running onto Ander Herrera's pass in the 29th minute.
Mkhitaryan, 27, settled the game with his first Premier League goal, running onto Ander Herrera's pass in the 29th minute and powering a shot past goalkeeper Hugo Lloris.
But the Armenian left the field strapped to a stretcher with six minutes remaining after his left ankle appeared to roll as he was caught by visiting left-back Danny Rose.
"We don't think it's a serious injury," Mourinho said. "We don't think maybe Henrikh can play for a week. I was afraid it was serious, but it looks like it's not.
"Maximum a couple of weeks. It doesn't look like something that is very dangerous or difficult.
"Maybe one week out and it is better to be fresh for Boxing Day."
Mkhitaryan has scored two goals in two games, having opened his United account against Zorya Luhansk in the Europa League on Thursday, and looks to be on his way to becoming a guaranteed starter.
"He brings everything I want an attacker to play. I am happy with that from Henrikh," Mourinho added.
"Not just the way he's creating, but his defensive participation in the team. He crossed the line to be a better player. He crossed the line mentally, that others have to cross.
"I want more from some of my attacking players. The two moments of recovering the ball are crucial moments. Mkhitaryan was superb on that."
United's win, in which Paul Pogba also hit the bar, ended a run of four successive league draws at Old Trafford and took them to within six points of the Champions League places.
Spurs are three points above them in fifth place, but now trail leaders Chelsea by 10 points, jeopardising their chances of repeating last season's title tilt.
"It was an even game, very competitive," said Spurs manager Mauricio Pochettino, whose side host Hull City in the league on Wednesday.
"I think a little bit disappointed, but not disappointed with our performance. We competed here, in a difficult game.
"Now we try to improve and need to be ready on Wednesday for a difficult game."