Perisic set up Nikola Kalinic before scoring a superb late winner in Bordeaux to send Croatia into the last 16 as Group D winners.
Double-defending champions Spain looked sluggish against a Croatia team that showed five changes from their 2-2 draw against Czech Republic.
"What can I say this was a really special evening for us," Perisic said. "I have to congratulate all my team-mates and our supporters we did a really good job in the group stage but this is just the beginning.
"If we lose the next match it will mean we didn't do anything important so we have to just focus on the next match and stay level and fight again like we did in the first three matches.
"This is the only way we can go far and to achieve a better result than Croatia in 1998 [third-place finish at World Cup]."
Cacic hailed the result's significance to Croatia, and was delighted to see some of his fringe players making an impact, with Kalinic impressing in place of the injured Mario Mandzukic.
"This was an important result for us. We won against the European champion," Cacic said.
"We won against a really strong team but it's not only about this match, we played the other two matches really strong team
"We will play against any team and we will play our game and be successful.
"Today I thought I can change five players and this is an important match for the next round so I want to play the next match with fresh players and test some really young players.
"Some young guys tonight, this game is really important. We wanted to be first after the groups and we achieved this. In all three matches we were really aggressive we were playing to win every match.
"We could not have done this without our great support, all these people who come from Croatia, we have to stay and fight and we want to make every single Croatian happy.
"I'm aware that winning against Spain - it's a huge achievement. It's big success and now we can be self-confident. You compare these players how strong we are to everybody but now we cannot think which way is better or to play against favourites or to play against not so strong teams.
"The draw is important but on the other side you have to win every game. Spain and Italy, this tells you in the last eight there will be no Spain or no Italy so maybe we have an easier path to the end but we just have to go step by step as I was saying after the first two matches."