Some supporters whistled at the team and new coach Jose Mourinho after a goalless first half at the Bernabeu before Mourinho's Portuguese compatriot Ricardo Carvalho scored to give his boss a winning home debut.
"I would ask the Bernabeu public for patience because we are a young team," Spain goalkeeper Casillas said at a post-training news conference on Sunday.
"We'll move forward step by step and each day we'll set our sights higher," he added. "We'll work calmly and little by little you'll see the results."
Expectations are high that Real can end a trophy drought and the domination of arch-rivals Barcelona after Mourinho joined from Inter Milan following a treble-winning season in Italy.
The Real faithful demand to be entertained, especially at home, and Mourinho and the players will have another chance to shine in Wednesday's Champions League match against Ajax Amsterdam.
Casillas said it would be dangerous for the Real players to let themselves be distracted by Barca's shock 2-0 home defeat to promoted Hercules on Saturday, their first league reverse at the Nou Camp since May 2009.
"We need to focus on our next match and our goal is to consolidate a group of players and a squad so that the machinery starts to function," the World Cup winner said.
"The team is more focused now and that's thanks to our all-round effort.
"We are solid and strong and we hope to stay that way. It's an important step forward that we have not conceded in our two league matches."
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