In wet conditions, United were laboured and their build-up play was slow as they struggled to break down their hosts, but Rashford provided a late spark after replacing Juan Mata, and converted Wayne Rooney's cross in the 92nd minute to crown a lively cameo.
Both sides sought to find their range during the first half as Robert Snodgrass sent a testing free kick just wide of David De Gea's goal in the 24th minute, while Eldin Jakupovic was forced into a low save from a Juan Mata set piece -- the first of a series of crucial stops.
World-record signing Paul Pogba was well shackled by the home side's disciplined midfield, although Zlatan Ibrahimovic went close only to be flagged offside after backheeling into the side netting.
David Meyler was fortunate not to be punished after raising an elbow to deflect a shot in the Hull box before Rashford's late intervention.