Jack Butland's failure to hold onto a loose ball allowed Alisson to give the South Americans an early lead before a slip by Liam Moore led to Lucas Silva doubling the lead.
Despite James Ward-Prowse's late free kick the Three Lions are now huge outsiders to reach Sunday's final, and former Middlesbrough boss Southgate called on his side to learn from their mistakes.
"We're disappointed, mainly because we felt we were equal to what Brazil had to offer. We felt we could come and win the game and during the game we showed that," he told BT Sport.
"We dealt with the occasion - it's the toughest test we've had - and I thought generally speaking we came through that very well.
"One or two looked nervous and the biggest disappointment is the goals we've conceded. They've not created anything brilliant and it's us that's scored a fantastic goal while conceding two poor ones.
"It was a tight game and we knew there would be little to choose between the teams.
"Those games hinge on a moment of magic or a mistake. We made two which is uncharacteristic for us but sometimes the tension of a game like that forces those errors.
"Experience-wise, our lads should think there's nothing to fear going forward."
The one bright moment on an otherwise poor night for England was Ward-Prowse's magnificent goal, with the Southampton man curling an effort into the top corner from the corner of the box.
And Southgate revealed he wasn't surprised to see the ball nestle in the back of the net, such is the 19-year-old's talent.
"It's no surprise that first of all he saw the opportunity and then to deliver the quality he did. That's what you get for hours and hours of practice, which is what he does.
"I thought he and Nathaniel Chalobah in the centre of midfield were outstanding."
England must now avoid defeat against South Korea on Wednesday to keep their hopes of claiming a place in the third-placed play-off alive.