Mkhitaryan converted Ander Herrera's through ball just before the half hour to end a frustrating run of four successive home draws and give United just a second win in nine league matches.
It was the Armenian playmaker's second goal in two games, after he opened his account for the club against Zorya Luhansk on Thursday.
He exited on a stretcher after turning his ankle in a late challenge by Danny Rose, but Mourinho said: "We don't think it's a serious injury. Maximum a couple of weeks."
Mourinho's team remain 13 points below leaders Chelsea, but closed to within three points of fifth-place Spurs and six of Manchester City, who occupy the fourth and final Champions League berth.
United once again failed to convert their dominance into goals, but unlike in their recent 1-1 draws against Arsenal, West Ham United and Everton, they were not made to pay.
"After so many matches playing well but not winning, we needed more the points than the performance," Mourinho said.
"But I knew against Tottenham it's difficult to get a result without playing well. But we played very well."
Initially kept on the sidelines by Mourinho following his arrival from Borussia Dortmund, Mkhitaryan has become something of a lucky charm for the Portuguese, who will be relieved the injury is not grave.
Defeat curbed Spurs' recent revival, which had seen them beat Swansea City 5-0 and secure a Europa League place with a 3-1 win over CSKA Moscow.
Mauricio Pochettino's side remain without an away league win since September 24.
"It was a disappointing result, but if we analyse the game I think it was a very competitive game and we need to take positive things from that," Pochettino said.
Pogba smacks bar
Tottenham centre-back Toby Alderweireld made his first start in eight weeks after a knee injury and he and his defensive colleagues quickly found themselves under examination.
It took United less than two minutes to carve Spurs open, Zlatan Ibrahimovic teeing up Paul Pogba for a left-foot shot that was parried by his France team-mate Hugo Lloris.
Mkhitaryan saw an effort blocked by Jan Vertonghen, Herrera gave Lloris some catching practice with a dipping volley and Michael Carrick whistled a shot over from range.
United goalkeeper David de Gea had to deal with a zippy free-kick from Christian Eriksen, but the hosts looked the more likely scorers and duly went ahead in the 29th minute.
Harry Kane played the ball straight to Herrera, whose inviting pass between the Spurs centre-backs freed Mkhitaryan to run through and smash a rising shot past Lloris.
The Spurs goalkeeper was called into action again shortly after, dropping to his right and saving from Ibrahimovic.
At the other end, Son Heung-Min cut in from the right and let fly with a left-foot effort that de Gea had to tip over, but the spring was in United's step as the teams went off at half-time.
Their failure to take their chances left them vulnerable to the threat of a counter-punch, however, and there were nervous moments early in the second half as Tottenham went on the attack.
De Gea boxed away a shot from Eriksen before Victor Wanyama gave the hosts a big let-off, completely miscuing his header after Eriksen's free-kick found him unmarked six yards from goal.
Seeking to soothe the home fans' frayed nerves, Pogba took it upon himself to try to extend United's lead.
The France midfielder left the Spurs crossbar shuddering with an arcing 25-yard free-kick and then stung Lloris's palms with a crisp shot following a one-two with Herrera.
De Gea had to claw away another Eriksen free-kick, but despite Mkhitaryan's premature 84th-minute exit, United saw the game out.