Hakim Ziyech’s injury-time goal sealed Chelsea’s win as they scraped through to the FA Cup semi-finals.
They took the lead courtesy of a first-half own goal by Oliver Norwood.
It came after Sheffield United failed to deal with Mason Mount’s right-wing corner and Ben Chilwell drove the ball towards goal.
Norwood attempted to block but inadvertently diverted the ball into his own net.
Christian Pulisic could have doubled the lead but was twice unable to find a way past keeper Aaron Ramsdale after getting through on goal.
The American showed great anticipation for the first chance, pouncing after Norwood lost possession, and for the second he cleverly dodged two challenges, but on both occasions Ramsdale stood firm.
United caused Chelsea major problems in the second half and created a number of chances.
The best of them fell to David McGoldrick, who contrived to head wide from close range after being found by John Lundstram’s right-wing cross.
Having struggled for much of the second half, Chelsea scored again right at the end when substitute Ziyech controlled Chilwell’s cross and fired home.
They remain unbeaten under Thomas Tuchel and have still not conceded a goal at home since the German took over as Blues boss.
Chelsea: Kepa, Zouma, Christensen (Azpilicueta 63), Chilwell, Hudson-Odoi (Havertz 72), Gilmour (Ziyech 72), Kovacic, Emerson, Mount, Pulisic (Pulisic 83), Giroud (James 63).
Subs not used: Caballero, Rudiger, Alonso, Werner.