Gareth Southgate's side enjoyed a dream start, taking the lead in the fifth minute courtesy of a wonderful Kieran Trippier free-kick.
"Until they scored [the equaliser] we were always under cosh", Kane said.
"And like I said, they should be more humble and respect more opponents".
Suddenly the massed ranks of red-and-white-clad Croatia fans behind Pickford's goal started to believe and shortly afterwards Perisic hit the post but England somehow clung on to take the game into extra-time.
Dead-ball situations once again played their part but were not the deciding factor for England and ultimately the "Love Train", as Glenn Hoddle has labelled it, was derailed. "We dominated the game physically, mentally, in all aspects". We worked so hard.More news: Serena beats Giorgi in three sets, makes Wimbledon semis
"We've left everything out there, you can look at the goals and the chances, but I don't think over the six or seven weeks we've been together we could've given any more". Croatia is the first team to win three straight extra-time matches in the World Cup.
After the match, England captain Harry Kane said the defeat would "hurt for a long time", although he reiterated that England had exceeded expectations by making it to the semi-final.
What is not in doubt is their quality.
For English soccer fans used to painful World Cup exits, the semifinal loss hurt.
As well as scoring, Perisic also hit the post for the second straight game and set up Mandzukic's victor, heading a poor clearance toward the Juventus striker to score.More news: Doubts if Saudis can replace lost Iranian barrels
"English journalists, pundits from television, they underestimated Croatia tonight and that was a huge mistake", Modric told ITV.
About 10 minutes before Mandzukic scored, defender Sime Vrsaljko headed a shot on target off the goal line to keep the score tied.
Vrsaljko, who plays for Atletico Madrid, had struggled to be ready for the match after playing in two previous extra-time matches, but he found the strength to lift his exhausted, battered team to a comeback win.
Mario Mandzukic put Croatia ahead after scoring in Moscow's Luzhniki stadium, giving the Balkan nation its first ever place in the finals in history, a result that surprised even some fans.
Croatia reached the first World Cup final in their young history on Wednesday, prolonging England's decades of pain as they set up a decider against France with a 2-1 win.More news: Top U.S. Senate Democrat vows to fight Trump's Supreme Court pick