"First half we were really good and we had chances," Southgate told ITV Sport. "We lost our way a bit after their goal. Possibly at that moment we were hanging on a bit rather than trying to control the game.
"We got the control back in extra time and showed composure. Knockout football is fine margins. When you have spells in a game against a quality side you have to make them count. We needed a second goal.
"I think we were looking at how we might stem the tide but there weren't clear changes we could make to change it. We just needed to show more composure. We stopped playing for a period of 20 minutes in the second half."
Up next for England is a third-place playoff encounter with Belgium in St Petersburg on Saturday.