Chelsea remained firmly on course for a domestic Double as N'Golo Kante's second-half winner settled a stormy FA Cup quarter-final meeting with Manchester United at Stamford Bridge.
United manager Jose Mourinho was involved in touchline clashes with opposite number Antonio Conte and was verbally abused by Chelsea fans at the scene of many of his triumphs, including three titles.
The Portuguese was furious when midfielder Ander Herrera was sent off 10 minutes before half-time after a second foul on Eden Hazard, and the managers were kept apart moments later after Marcos Alonso tumbled to the floor after being brought down.
Kante's low 51st-minute drive finally beat defiant United keeper David de Gea, who saved superbly from Hazard and Gary Cahill before the break to keep Mourinho's side in contention before Chelsea made the breakthrough.
Marcus Rashford, who came off his sick bed to play - with Zlatan Ibrahimovic suspended, and Wayne Rooney and Anthony Martial injured, created United's best chance for himself but Chelsea keeper Thibaut Courtois saved with his feet to set up a semi-final against Tottenham at Wembley.