"We were playing against world champions and were trying to neutralise their game," Bilic said after the match, which ended 1-0 thanks to an 88th-minute winner from Spain substitute Jesus Navas.
"I think they had a hard time," he said, underlining that Croatia had battled hard. "Some teams can beat Spain, but it's not easy. We knew we'd have two or three opportunities. It's not easy when you play against Spain. You have to score when you have the chance."
Having failed to progress after Italy sealed the other quarter-final berth in Group C by beating Republic of Ireland 2-0, Croatia's hopes of a return to their 1990s glory, when Bilic was a player, were dashed.
Bilic said, however, that a second consecutive European title was not certain for Spain. "We all know they are world and European champions, they want to win. They want trophies. They are the best team in the world," he said.
"We were close, it's true, we could have won tonight and we could have stopped them tonight. I don't see them as big favourites. There are loads of teams that maybe don't have the same class, but they are hungry and aggressive," he insisted.