David Villa scored the only goal of the game to set up a last eight showdown with Paraguay and Casillas believes the South Americans will provide another tough test for his side as they look to take another step towards World Cup glory.
"We competed well. We continued playing well after the break and we have achieved a fair victory," he said. "The team was phenomenal at the back and have not allowed Cristiano [Ronaldo] to create danger.
"It is a narrow margin, we always want to win by more, but in this World Cup we have learned to handle this score line better.
"We are meeting our objectives step by step. The team did well in the first and in the second half. We knew it would be difficult to beat Portugal because they are a very tough team, but Spain have been true to our style of play. The result is a credit to everyone."
The Real Madrid shot stopper also declared that his team must show respect for the Paraguayans, despite noting that the European champions are in good shape going into the quarter-finals.
"There is no easy opponent, there are teams that at first seemed small but we know that we must have respect for them," added Casillas.
"Sunday is a game to enjoy and win. Paraguayare very competitive, like Chile they don't lose their composure. It's not going to be easy to get into the next round."