Cameron Archer beat Gabriel Osho for pace on the right flank and slotted into the bottom corner before James McAtee converted from the penalty spot, as the Blades led by two league goals for the first time this season.
Carlton Morris got one back for Luton, scoring a spot-kick shortly after the break.
Vini Souza restored United's two-goal cushion with a well-struck finish 18 minutes from time.
Chris Wilder's side remain bottom of the league, seven points from safety, while Luton are just above the drop zone in 17th position.
Were you at the match or did you follow it from home?