Jon Dagur Thorsteinsson struck the winner for Iceland in England's final tune-up game before the tournament.
Asked if England fans should be worried, Southgate said: "Naturally after the way we played tonight they will be but that's not been the performances we've put on in the last 12 months. It's actually a good performance in terms of focusing the mind on what needs to happen because you can't be loose, it's not just about talent to go and win football matches.
"You have to be really well-organised, you have to be disciplined without the ball, and when you're going to press you have to be co-ordinated because otherwise you can be picked off and that's what happened at times tonight."
On reasons for the performance, he continued: "I think there were a lot of factors. I think the biggest one was that a lot of players physically needed the game, players that were short of minutes coming into the camp but even the players who have played a lot this season and haven't now played for a fortnight. I think getting the minutes under the belt was important tonight as well."
On whether a week is long enough to correct England's issues, Southgate concluded: "It's enough to get us in a better place than we were tonight and improve the performance of tonight because we know that has to be the case. We know the bits that weren't good enough but we also have to stay calm because we've been through nights like this before.
"The bigger problem is that if you don't know what the issues are, but the issues we can see very clearly tonight and we have a chance to address them over the next week."