The Blues will resume a tempestuous rivalry with the Yorkshire side, while the Gunners travel to the League One outfit and Paul Lambert returns to Carrow Road in the last eight
The Blues won 5-4 after extra-time at Stamford Bridge on Wednesday evening, after Eden Hazard netted a 94th minute penalty to deny the Red Devils their place in the last eight.
Arsenal's reward for their own 120-minute win over Reading is a trip to the lowest ranked side left in the competition, Bradford after the Bantams defeated Wigan on penalties after a 0-0 draw on Tuesday night.
Middlesbrough's reward for their win over local rivals Sunderland is another trip to a Premier League ground, although with a much further distance to travel as Tony Mowbray's Championship high-fliers drew Swansea.
The Swans defeated Liverpool at Anfield in their fourth-round clash, knocking former boss Brendan Rodgers out of the competition and reaching the last eight for the very first time.
The final tie sees Paul Lambert return to Carrow Road for the first time in the only all-Premier League clash, with Aston Villa paired with Norwich City.
Draw in full:
Leeds v Chelsea
Swansea v Middlesbrough
Norwich v Aston Villa
Bradford v Arsenal