The two teams - along with Group C rivals Italy, who were facing the Republic of Ireland in Poznan at the same time - went into the closing minutes of last night's action with their progression still in the balance.
Jesus Navas' simple finish settled Spanish nerves though, while Mario Balotelli secured Italy a 2-0 win over the Irish to ensure the group big guns marched on.
Casillas said: "I think it was a very tense match, but that is normal because it was important to get through to the next round.
"Croatia were well prepared to face us. They made it very difficult for us to create chances, they defended very well. In the end we were lucky, scoring late on. It felt like a weight off our shoulders."
Goalscorer Navas added: "It's great to qualify this way, especially if you win. We added more points to our tally and finished first in our group. We are very happy about it.
"Croatia were great opponents but we were able to fight our way to this victory. We always go for the win. That's what we believe in, that's what we play for."
France, England or co-hosts Ukraine await in the last eight and Navas said: "Whoever we have to face, we will stick to our game. I am really looking forward to whoever we will have to play."
Croatia had chances to snatch victory - not least when Ivan Rakitic's diving header forced a sharp save from Casillas, who said: "I was lucky in one situation that could have put us in trouble."
Rakitic added: "I did not manage to take the chance that came my way. I think I will probably have sleepless nights thinking about that.
"It was obvious that we gave everything on the pitch, we gave it our best shot. I can honestly say that my soul was left on that pitch, and I think I need a few people to carry me off the pitch because I am literally dead."
Team-mate Danijel Pranjic added: "As long as the match was still scoreless we had a chance. If Rakitic had scored, who knows how this match could have ended.
"If you can make Spain nervous until the last minute of the match about getting through, I think that is an outstanding performance."