Alas, a second-half goal by Croatia sent the game into extra time and Mario Mandzukic's extra time victor put the Croatians into the final against France and relegated England to the third-place game.
Perisic almost scored a second goal just minutes later, as he latched onto the ball following a defensive lapse before striking the post with his shot. Trippier struck a sweet effort over the Croatia wall in the fifth minute, guiding it perfectly past goalkeeper Danijel Subasic and into the top scorer.
Mandzukic had Croatia's best chance at the end of the first additional period, but Pickford blocked his shot from close range.
The best chance for a second England goal was wasted by an unmarked Jesse Lingard when he curled wide when a goal looked certain. A lovely header by Harry Maguire in the first half was all they needed to win.
England paid the price for those misses when Perisic showed great determination to get in front of Kyle Walker to meet a Sime Vrsaljko cross and though his boot was high and he threw an anxious glance at the referee, Walker was stooping and there were few complaints.
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Federer was completely out of sorts by now and his uncharacteristic struggles on his forehand didn't improve in the fourth set. Federer seemed to be heading for a routine victory with match point to complete a straights set win in the third.
But Croatia, having won on penalties against Denmark and Russian Federation in the last two rounds, saw the game out, dousing the World Cup dream of Gareth Southgate's side that had gone from the improbable to the possible over the last month in Russian Federation. Perisic shifted the ball on to his left foot and smashed a shot across Pickford.
Elspeth Patient, 34, who is originally from Milton Keynes but now lives in Los Angeles, said: "It's been awesome, it's been such an incredible time for England".
Neither team came to Russian Federation being heralded as favourites but they have produced performances throughout the competition which have shown they were underestimated by the pundits. England won a penalty shootout against Colombia in the round of 16, but beat Sweden 2-0 in the quarterfinals.
World Cup Daily: Croatia's dream goes on while England wake up.
England had been the ones hoping to make history, in what was their first appearance in the semifinals in 28 years.
Mario Mandzukic was Croatia's match-winner, swivelling to power home the decisive goal in the 109th minute. This time, the Croatians have only recorded a few shots at Jordan Pickford's goal from outside the penalty area, a lot of them off target. His chief task on Wednesday will be to limit the space afforded to the likes of Raheem Sterling, Jesse Lingard and Kieran Trippier, who have opened up rival defences in Russian Federation with their clever runs and interplay.