Ex-Chelsea striker Romelu Lukaku netted a second half brace to ensure the Toffees' advancement to the cup quarter-final stage.
The Belgian international opened the scoring in the 77th minute thanks to a sublime run from the left flank, cutting in and beating four Chelsea defenders before slotting home from inside the box.
Lukaku then completed his brace five minutes later when he confidently stroked the ball past Thibout Courtois after a lovely lay-off.
Diego Costa then recieved a bizarre red card for appearing to have bitten Gareth Barry in the 85th minute, before Barry himself recieved his marching orders in the 88th minte for a second bookable offence.
The two goals ultimately ended The Blues' season as the loss saw them out of contention for the last piece of silverware on offer this season, with the league title way out of their reach.
Crystal Palace beat Reading 2-0 in the first quarter-final played on Friday evening, while Arsenal face Watford and Manchester United host West Ham in the remaining last eight fixtures on Sunday.
