The Croatians finished third in Group C on four points, with Spain (seven) and Italy (five) reaching the quarter-finals.
A late goal from Jesus Navas secured victory for Spain although the European and world champions were pushed all the way by Croatia.
Rakitic told reporters he would now have "several sleepless nights" because of the miss.
"We came here to go all the way and we failed because we lacked that tiny bit of luck that makes all the difference," Bilic said.
"We knew we would have two or three chances and we had the one with Rakitic and when you get a chance like that you have to take it."
Bilic said Croatia should have been awarded two penalties but Spain enjoyed "permanent help from the referee in those little details".
Croatia would now, Bilic said, head home "with our heads held high".
"This was a really difficult group and we are proud of what we did," he said.
"We can go back home with our heads held high. To play against Spain is really tough. They are such a tactical team."
"I don't see them as the really big favourites," he said. "There are quite a few teams that maybe have more pace and are hungry to win and maybe more aggressive on the pitch than Spain."
"I didn't expect this would be my last match," he lamented.
"I was sure we would go through because we are a good team. My players are just phenomenal.
"I am not going to make a big speech, it's not my style."
