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.
Pedro stepped off the bench to send Chelsea into the FA Cup semi-finals after they beat Leicester 2-1 at the King Power Stadium on Sunday evening.
RB Leipzig came from behind at the Red Bull Arena to end a run of four winless games and defeat Bundesliga leaders Bayern Munich 2-1 on Sunday.