A match that became a long battle between two teams displaying warrior spirit began with a Kieran Trippier free kick within the first five minutes.
However, Croatia stepped things up a gear in the second period, with Ivan Perisic volleying home from close range to send the game in to extra, before Mario Mandzukic broke English hearts in the 109th minute when he took advantage of some exhausted defending to slot a tidy finish past Jordan Pickford.
Asked about Saturday's match against Belgium, who beat England 1-0 in a somewhat meaningless group stage match on June 28, Southgate said: "The honest thing is it's not a game any team wants to play in".
Some England fans emerged from the Luzhniki Stadium in tears and being hugged by others while jubilant Croatian supporters celebrated around them.
The crowd fell a little flat as the game went into extra time, with many perhaps fearing the prospect of penalties and the memory of the last time England made it so far at Italia 90 - a 4-3 penalty shootout loss to West Germany.
"Semi-finals, they've done what they had to do, they took us to the semi-finals". We're going to celebrate, to rest and then we will prepare for France. We did not underestimate them.
"I just feel so disappointed".
They had only previously been in the World Cup semi-finals in 1966, when they won the title on home soil, and 1990 so Southgate said he was trying to balance his pride in his young team with remembering how rare such chances are.
"We have to keep the same standard", Walker said.
"We were motivated and what I told these players before the game was "there's no pressure, no nerves, don't lose your head - you've reached the semi-final, be proud and enjoy your football".
"They played their best, it obviously wasn't to be".