England suffered a 3-2 defeat to Italy in the shootout, after the match finished 1-1, with Rashford striking the post before Italy goalkeeper Gianluigi Donnarumma saved from Sancho and Bukayo Saka.
England have won just 22 per cent (two out of nine) of their major tournament shootouts, the lowest ratio of any European nation to have been involved in three or more.
When asked if Rashford and Sancho, both brought off the bench in extra-time a few minutes before the shootout, might have been "cold", Southgate said the decision was one fraught with danger.
"That is the risk you run but they have been the best in the lead in, and to throw all those attacking players on you have to do it late," he told ITV.
"You have got to have balance to the team. You can't just throw on attacking players earlier, or you lose your shape and don't have a foothold in the game.
"It was a gamble but if we gambled earlier in the game, we maybe lose the game in extra time. I chose the guys to take the kicks."
Southgate revealed England's order for the penalty kicks had already been decided in advance of the Wembley final, and said the decision to let Saka take one was his alone.
He added: "It's my decision to give him that penalty so that is totally my responsibility. Not his.
"The same with Marcus or Jadon. We work together, worked through them in training, that was the order that we came to.
"I said afterwards that nobody was on their own in that situation [missing a penalty].
"We decided to make the changes late in the game and we lose together as a team. The players have been tight throughout and that's how it will stay."
Southgate said he was proud of England's overall display in the tournament although felt they were poor in possession in the final.
"In the end we were not quite able to see the game through in normal time and Italy showed the outstanding team they are with 30 plus games unbeaten," he explained.
"Our players have to be proud of themselves. Every one of them has been exceptional. First time we have to a final and we are very disappointed not go on a step further.
"Italy have some outstanding players but we didn't keep the ball well enough in that initial period in the second half. We changed the shape to get more of a grip. It was a lack of composure in possession which turned the game."
England will now switch their focus to next year's World Cup in Qatar, although Southgate needs time to let the wounds heal from this defeat.
"It's hard to reflect at this moment because the disappointment is enormous for all us," he said.
"The players have done us proud. The way the nation have got behind us. I know tonight has burst the balloon. But I hope everyone remembers what this group has given them.
"We have given everyone some fantastic nights and we wanted to give them one more and came close to having done that. It's hard for me to put that into words at the moment.
"At this moment it is hard to look that far ahead. This was a wonderful opportunity and we need let that sink in before thinking about Qatar."