Carlo Ancelotti's men are now on a four-game winning streak in the Spanish top flight after Cristiano Ronaldo and Karim Benzema wrapped up the points at the Santiago Bernabeu on Sunday.
Ronaldo scored a hat-trick, taking his tally for the season to 13 goals from just six league outings, while Benzema added a brace as Real's forward ranks combined to leave Bilbao helpless.
However, Casillas was more pleased with the clean sheet and believes the team's all-round showing bodes well as they look to make up the four-point gap to leaders Barcelona.
"The result could have gone differently but we were good across the park. We still have to catch up with the league leaders," said the experienced goalkeeper.
"The important thing was to continue with our form, slowly but surely we'll make up the difference.
"We had been conceding a lot of goals so this is an important result. The team was full of confidence and we were solid throughout. We players at the back feel the support from the forward players.
"We had a fruitful night in front of goal without conceding, which is what the fans want."
Casillas also paid tribute to Ronaldo, who netted his 22nd Liga hat-trick to equal the Spanish top-flight record held jointly by Telmo Zarra and Alfredo Di Stefano.
"We all know how great a player he is, his streak of form and his numbers are there for all to see," added the 33-year-old.
"He has broken all the records in terms of goalscoring and assists. We are delighted for him.
"The greatest thing is that the fans are behind the team, that is the main thing and it will make us play at our best."
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