England had defied all expectations in reaching the last four and led 1-0 at half-time thanks to Kieran Trippier’s fifth-minute free-kick.
But Croatia rallied in the second half, levelling the match through Ivan Perisic before Mario Mandzukic struck in the second half of extra time.
Dignified in defeat as he has been in victory, Southgate said: “It’s really painful at the moment but I’m proud of the players, the way we played and the strides they’ve taken.
It’s very difficult to say anything that will make them feel better in this moment.
“Tonight, in the end, we weren’t quite there but the team will be stronger for that. “The next two days there will be a lot more positives but tonight is so hard to take...we were so close to what was almost unimaginable at the start.”
One of those reflections might be that England should perhaps have had the game wrapped up by half-time with Harry Kane and Jesse Lingard missing glorious chances to build a clear adavantage.
Southgate added: “You’re ahead in a huge game and nobody knows how you’re going to react. “We lost a little bit of control with the ball and started to go a bit longer.
“There were a few waves of attack we didn’t quite defend in the composed manner we have but great credit to the players because they recovered that in extra-time and I felt took control of the game again.
“On good chances we’ve probably had as many as Croatia have. “I felt there was a period of 20 minutes in the second half when we stopped playing and lost that composure. “And it’s a blow to concede as late as we did.”
Southgate also praised the travelling fans and those back home, and offered them hope for a bright future.
“We know the team we brought was the most inexperienced in the competition but we gain that experience from matches in this tournament and performing under pressure,” he said.
“You don’t get that reaction (of the supporters in Moscow after the game) if you haven’t played as well as you have. We’ve left everything out there. We will look at the goals and chances but I don’t think over the six or seven weeks together we could have given more than we have.”