Ole Gunnar Solskjaer's team went into the game riding a nine-match away winning streak in the league, but went down inside of five minutes when second-choice keeper Dean Henderson lost the ball in front of his own net and David McGoldrick pounced to fire Sheffield United in front.
The away winning streak aside, it was the sixth consecutive match that Man United conceded the first goal while playing on the road, having come back to win the previous five.
Man United spent most of the next 20 minutes in the hosts' end of the field and got back on level terms when Rashford settled a pinpoint long ball from Victor Lindelof and blasted his next touch past Sheffield keeper Aaron Ramsdale.
Soon after, Anthony Martial put Man United in front from a brilliant Paul Pogba pass that Ramsdale misplayed and the France international poked home to give his team a 2-1 lead they would take into half-time.
Solskjaer said after the match: "It could have been more comfortable. We were a little sloppy at 3-1 up. The corner that led to the second goal wasn't a corner but when it goes in it was really difficult because they are a handful."
Martial set Rashford up for Man United's third from a counter-attack just after the restart, teeing up his teammate in the centre of the penalty area where he dispatched his second goal on the night.
McGoldrick scored his second of the night for Sheffield United with a header following a corner to set up a nervy finish for Solskjaer's side, but they held on to see out the result and pick up all three points.
Manchester United are now up to sixth in the Premier League on 23 points from 12 matches and have a game in hand on all their top-four rivals ahead of them.
Asked about his players believing they could yet win the title, the Norwegian was cautious.
"We can't get our heads too far ahead, but I am happy the players might be thinking that way because that means recovery, focus is better and those little margins mean a lot," Solskjaer said.
Sheffield United are stuck at the bottom of the table with just a single point so far this campaign, setting a new record for the worst start for any top division team in England since league play began in 1888.
Despite the ignominious mark, Sheffield United boss Chris Wilder took heart from his side's performance.
"It has given myself and the players a belief that if we do play in the manner we played - especially out of possession - that there is something left in this season for us," he said.
"I will stick my head on the pillow tonight knowing the team has left everything out there."