Karim Benzema netted just before half-time to put Jose Mourinho's side 3-0 up in the tie, before Coentrao picked up a second yellow card for a handball in the 51st minute.
Valencia pulled a goal back from the ensuing free-kick when Tino Costa's low free-kick surprised Real's reserve keeper Antonio Adan.
Adan had to replace Casillas in the 15th minute after Alvaro Arbeloa accidentally kicked his hand while trying to clear a ball.
The injury to Spain's number one did not appear to be too serious as he was later seen in the dugout with ice on his hand.
Di Maria picked up a foolish red card two minutes from time after kicking out at Valencia defender Joao Pereira as the two tussled for a ball.
Real will face either holders Barcelona or Malaga in the last four. Barca visit the Rosaleda on Thursday with their tie level at 2-2.
On the other side of the draw, Alvaro Negredo scored twice to help Sevilla past nine-man Real Zaragoza with a 4-0 quarter-final second leg victory.