Norwood turned Ben Chilwell’s strike past his own goalkeeper midway through the first-half, but it was only in stoppage time at the end of the game that Hakim Ziyech crowned a 2-0 win to book Chelsea’s place in the last four.
Chelsea could have had a second earlier on through Christian Pulisic, but the American forward lost his footing and his shot was blocked by Aaron Ramsdale.
Sheffield United were rejuvenated after the break but lacked the killer instinct in front of goal and squandered a couple of clear chances, David McGoldrick guilty of somehow heading wide when gloriously placed.
The victory extended Thomas Tuchel’s unbeaten start in London with the German winning 10 matches and drawing four in all competitions since replacing Frank Lampard in January.
Tuchel’s side will discover who they face in the last four when the draw is made at halftime of the quarter-final between Leicester City and Manchester United later on Sunday. The semi-finals take place next month with the final on May 15.
Additional reporting by PA.