The home side led when Marouane Fellaini headed in Ander Herrera's left-wing cross at the far post.
Stoke equalised through Steven N'Zonzi's low shot, but they slipped to a third straight loss when Juan Mata's free-kick from the right sneaked in.
Stoke's Mame Biram Diouf was twice denied an injury-time equaliser.
The 26-year-old Stoke forward, who had an undistinguished spell at Old Trafford earlier in his career, saw United keeper David De Gea claw out his header before emergency full-back Ashley Young cleared another effort off the line seconds later.
Diouf was left mystified by his failure to earn a point for the visitors, shaking his head in disbelief as he slowly trudged off the pitch at the final whistle.
A draw would have been harsh on United, who controlled the game throughout but had to rely on a piece of fortune for the winner.
Mata's set-piece evaded every player waiting in the penalty area before finding the net, although team-mate Marcos Rojo claimed he got the final touch with his head.
United boss Louis van Gaal will not be too bothered who scored. The Dutchman's team continue to build momentum, despite failing to turn almost 65% of possession into more goals.
Skipper Wayne Rooney and Angel Di Maria were both missing from United's squad through injuries sustained against Hull on Saturday.
And for long periods, the home side missed the pair's attacking guile.
Stoke rarely looked like ending a dismal run of 11 successive Old Trafford defeats, though, until Diouf and substitute Marko Arnautovic went close in the final moments.
Old Trafford lost some of its famed 'fear factor' last season as United slumped to seven home Premier League defeats in a terrible first campaign following Sir Alex Ferguson's retirement.
That aura of invincibility appears to have returned under Van Gaal, whose team have taken 19 home points from a possible 21 since defeat by Swansea on the opening day.
Potters boss Mark Hughes indicated his side would not be afraid to "have a go" at United, but they rarely looked like carving open a makeshift home defence until stoppage time.
United controlled the tempo and deservedly led when Herrera was allowed too much time and space to pick out Fellaini, who outjumped Erik Pieters to powerfully nod in.
Hughes was furious with the space that his side allowed Herrera, but his demeanour changed when N'Zonzi calmly guided a 20-yard shot past De Gea with Stoke's first attempt on goal.
But then Mata's intervention moved United within two points of second-placed Manchester City, who travel to Sunderland on Wednesday, while 13th-placed Stoke remain four points above the relegation zone.
Manchester United manager Louis van Gaal:
"It was not our best performance. We were not aggressive enough in the first half, we didn't keep our position on the pitch. I was not pleased at half-time - and I told them (the players). The second half was better, we created more chances but we need to score more goals.
"Then Peter Crouch comes on, you know the long balls are coming and it is always difficult. But I'm pleased we held this result. The last few moments were not necessary, but it is always like that. We have struggled until the end in the Premier League. It is common.
"I'm happy with the result. Tomorrow we have a Christmas party and it's important we enjoy that."
Stoke manager Mark Hughes:
"On the balance of play, we were very much in the game, especially in the first half. It was a really good performance and we picked ourselves up from the disappointment of conceding early on.
"We are disappointed with the award of the second goal. We clearly think it is offside. It doesn't matter if he (Rojo) touches it, he is clearly offside and interfering with play.
"He was ahead of the ball in an offside position. The referee and assistant should have cancelled that out. We didn't allow that to affect us but the episodes at the end... it was unbelievable, demonstrating the sort of luck we are having at the moment."
(BBC)
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