Defender Victor Lindelof scored in the 58th minute as his goal proved to be the decider as Luton Town lost its 17th straight game at Old Trafford.
Lindelof smashed home from 16 yards after Luton failed to clear Marcus Rashford’s cross to grab United’s opener in the thrilling encounter.
The Red Devils , who have won two successive league games after stumbling to their worst start since 1962, provisionally climbed two spots in the standings into sixth place on 21 points after 12 games.
Luton Town, who have just one league victory in their debut season in the top flight, are 17th on the Premier League log currently.