Mourinho has found life tough this season with Madrid already 11 points behind arch-rivals Barcelona.
Fans jeered Mourinho's name during Tuesday's Copa win over Alcoyano.
"It (the fans' booing) may be unfair on Mourinho. He took on a new team with young players. Much of the success has been down to him, we have had some very good seasons recently," said Casillas.
"He is the best coach in the world and has shown this.
"Our trust in him is absolute. Mourinho is a coach who has given us much. Barcelona were playing well and went on to win titles and over those seasons defeated us. Last season the clear favourite was Barcelona.
"We hope to continue to regain ground in the league. We thought we might be closer going into the derby if we had won in Seville, but it cannot be and we will try to now make it a five-point gap with the derby.
"Right now, both teams are ahead of us. In recent years we have been fortunate to have gone unbeaten against Atletico, but that does not mean anything when we are looking for nothing but a win. The draw is worth nothing to us."