Southgate, one of Crawley's favourite sons, has been the pride of the nation in leading England to the final - their first in a major tournament since 1966.
But the adventure ended in heartbreak and tears as England lost 3-2 on penalties to Italy, with Marcus Rashford, Jadon Sancho and Bukayo Saka all missed spot kicks.
The shootout followed a 1-1 draw, with England unable to hold on to the early lead given them by Luke Shaw's goal.
Southgate told BBC TV: "Tonight is going to be extremely difficult for all of us, of course. You have to feel that disappointment because the opportunities to win trophies like this are so rare in your life. But when they reflect on what they have done, they should be incredibly proud of themselves.
"The players have been an absolute credit. They have given everything they possibly could. They have been an absolute joy to work with and they have gone further than we've gone for so long. But it is incredibly painful in that dressing room."
Of the penalty shootout defeat, Southgate said: "It is down to me. I decided on the penalty-takers based on what they have done in training. Nobody is on their own. We have won together as a team and it is on all of us together to not be able to win this game. In terms of penalties, it is my call and it totally rests with me.
"We were well prepared for penalties and we started it well but the guys weren't able to convert. But we are all together and the players have given the country some incredible moments. Chances to win trophies like this are so rare but when the players reflect they should be proud of themselves.
Asked about the positives he could take from England's campaign, Southgate said: "In a few days that will be more possible. But now the pain of a defeat is huge. We wanted to win and have not been able to do it."
Asked about the 2022 World Cup and England's prospects, he said: "I can't really talk about that tonight, it seems a million miles away. I need a bit of time to reflect."
Former England midfielder Frank Lampard, speaking on the BBC's coverage of the final, said Southgate would make sure the players who missed spot kicks were given support. Lampard said: "If there's one man who can put his arm around them and support them it's Southgate because he's been there. It's going to hurt. There is nothing you can say. They stepped up but football is so, so cruel."