Perisic scored for Croatia mid way through the second half to take it to extra time.
Croatia's Mario Mandzukic scores their second goal in their 2-1 win in their semifinal against England.
He said: 'I don't think people fully understood the players (previously).
Croatia had been desperately poor, a shadow of the team who carved Argentina apart in the group stage, but started to get a foothold in the game as the match rolled past the hour mark and Luka Modric's influence grew.
Having been knocked out of Euro 2016 by Iceland and failing to win a single match at the 2014 World Cup, optimism was not high heading into Russian Federation.
England's work in Russian Federation isn't yet done as on Saturday they will face Belgium, who were in their group, in the third place play-off.
Harry Kane shot weakly at Danijel Subasic when through, then forced the rebound against a post and Raheem Sterling was a constant thorn in the Croatia defence, though again his final ball was too often astray.
"We can hold our heads up high", said team Captain Harry Kane after the loss.
"Some players played with minor injuries with which they would not have played some other games". I thought they played well and made it hard for us.
Modric, whose team initially went 1-0 down to England thanks to a sublime free kick by Kieran Trippier in the fifth minute, accused the English press of showing a lack of respect to Croatia.
Some fans claim Croatia seemed to get away with a load of fouls, while England were immediately carded for lesser ones. When you have spells, you have to make them count.
Our boys did us proud during this World Cup, playing like a proper team and proper gents.
Kane looks set to claim the Golden Boot as the World Cup's leading scorer, but the Tottenham star rarely looked like adding to his six goals against Croatia. With just 9 minutes to go, Croatia tactfully slowed the game down and started ringing in the changes, just to kill time.
The thinking was that if all players had left the field of play, the opposing team would be allowed to kick-off with only the goalkeeper left to defend the goal.
"But we have our hearts, our pride and our players and that is what is really important to us".