La Roja topped Group C following their hard-fought 1-0 win, with Jesus Navas netting the winner three minutes from time to seal the three points.
And Casillas was happy to see his side progress, but acknowledged that Spain must deal with the pressure better if the team is to reclaim the trophy on July 1.
The goalkeeper told Telecinco: “We had to suffer, but it's normal. There's not a single weak rival, and everyone who plays against Spain give the 100 percent. We've been good taking advantage of the situation.“
The captain had to be in top form to keep a clean sheet, with Ivan Rakitic’s close-range header the biggest threat.
Casillas added: “Fortunately, I was running to the ball, [and] in the good place, and I could stop his header. I was feeling good tonight. And I left the pitch with happiness."
And the shot-stopper blamed the players’ nerves in what led to a difficult performance against Croatia, but feels that after advancing from the group the ‘do-or-die’ nature of the knockout round could now help Spain to perform better.
He added: “It was difficult because we went all nervous. With an isolated action we would have been out. Now we have got a life or death game, and I think that's not bad at all for Spain.
“We go step by step, and then we'll see where can we get.
"I don't think this is the same Spain than World Cup. But it's because everyone know us already. Now, when [Andres] Iniesta or [David] Silva gets the ball, they [have] got three players with them. Spain [have] got a marvellous touch, and everyone knows that."