Kieran Trippier fired England into a fifth-minute lead but Ivan Perisic forced extra-time, where Mario Mandzukic won the match for Croatia.
“At the moment we all feel the pain of the defeat,” England manager Southgate told FIFA.com. “Did we expect to be in this position? I don’t think realistically any of us did.
“But when you’ve got to this point and you’ve played in the way we have and played as well as we have in the first half you want to take those opportunities in life.
“The dressing room is a very difficult place at the moment. But what I will say is I’m remarkably proud of a group of players who have really advanced.
“I think the reaction of the supporters to them, compared to two years ago, tells them that, first and foremost, that experiences with England can be positive, the country are very proud of them and the way that they’ve played and there will in time be a lot of positives to take.”
England were not overawed by their first World Cup semi-final since 1990, dominating the first period of the Luzhniki Stadium.
But having failed to add a second to Trippier’s opener, England’s youthful side tired and succumbed to an experience Croatia team.
“I don’t know [if we deserved better],” Southgate added. “In the first half, we were really good and we maybe could’ve got another. At periods throughout the game we had chances.
“There was a spell in the second-half, they got the goal and their tails were up and we had to really weather the storm. We lost our way a little bit. Possibly, at that moment, that experience of being ahead in a game like this, we were hanging on a little bit rather than trying to control the game.
“They deserve great credit in extra time, and showed some more composure. You see the supporters at the end and their reaction and that shows what the players have given not just tonight but over the tournament.
“Knockout football is fine margins, if you have spells against a side of real quality then you have to make them count. We probably needed that second goal earlier in the game.”