The Canaries took Premier League champions Chelsea all the way to penalties in their FA Cup third-round replay at Stamford Bridge on Wednesday.
Those efforts seemed to have had an impact when James Wilson, on his full debut, put the Blades ahead in the sixth minute, turning the ball home following a corner.
The visitors doubled their lead in the 68th minute when Clayton Donaldson capitalised on an under-hit Alex Tettey pass-back to slot past goalkeeper Angus Gunn.
Preston are reportedly weighing up whether or not to make a fourth offer for Peterborough United left-back Andrew Hughes.
James Maddison has completed his move from Norwich City to Leicester City for an undisclosed fee.
West Ham United are reportedly weighing up a summer bid for Amiens forward Moussa Konate.