A poor match in Lens saw no shots on target until the 117th minute, when Cristiano Ronaldo's strike was palmed into the path of Ricardo Quaresma, who headed in the winner.
Croatia had emerged as one of the tournament favourites after stunning two-time defending champions Spain to top Group D, but the Balkan nation will now head home as Portugal prepare to face Poland in the quarter-finals.
"What went wrong? Everything was perfect, except we didn't score. We were not successful in attack, we expected Portugal to play on the counter," Cacic told reporters.
"We controlled the match for 120 minutes, we didn't concede any chances except the last one, when they punished our mistake. That's football. The better team often doesn't win, and this was the case tonight.
"I'm very sorry for it. Both teams played very well but we were dominant, played more aggressively but they were disciplined, defended really well. What can I say? They scored on 117 minutes and I can only congratulate them. They kept us far from the goal and they scored on the counter because we were trying to score until the end.
"Of course I'm really sad. The draw was perfect for us, we knew it would be a really open path if we beat Portugal. But that's tournament football. Portugal are a strong team, you saw that tonight. I can only congratulate my players again. But they have high quality and experience.
"Portugal scored the goal and we didn't, that's it. They had the only chance so we just have to look to the future and start to think about the World Cup in Russia."