Despite taking four points from their opening two games, England's performances in a 1-0 win over Serbia and 1-1 draw with Denmark have been fiercely criticised for failing to deliver on their tag as pre-tournament favourites.
"We've always known we are one of the team who can win the tournament," Southgate told reporters at his pre-match press conference.
"We haven't changed our objectives. We know to do that, we have to show a different level to what we have so far."
Under Southgate, England have reached a final, semi-final and quarter-final in the last three major tournaments.
However, that has not sheltered him from criticism back home after a stuttering start to what many believed would be the tournament to end a 58-year wait for glory.
"I don't think the performances over the last two matches have reflected how we've played over the last couple of years. We've had a short period here where we're not functioning at the level we want to," added Southgate.
"Our performance will have to talk for itself. It's important to answer the questions as well as I can but I will be judged on the way the team play. I understand that and I'm comfortable with that."
Slovenia out to 'shock the world'
Slovenia have drawn their opening two games and realistically need to take something from the match in Cologne to have any chance of reaching the last 16.
England's Declan Rice said his team are wary of the threat posed by Slovenia striker Benjamin Sesko, who had been linked with a move to join the midfielder at Arsenal, before recently extending his contract with German club RB Leipzig.
"I have watched Slovenia and they have been really good, they have strong players who are really fast. The main talking point is Sesko -- he's been a massive target for clubs round Europe," said Rice.
"They are going to come at us and try to win and shock the world."
Southgate's captain Harry Kane hit back at some of the critics, including former England striker Gary Lineker, for what he perceived as an over-the-top reaction to the Three Lions' struggles.
Kane reminded the likes of Lineker and former skipper Alan Shearer that they also had not won anything while playing for their country.
By contrast, Southgate has shut himself off from media coverage during the tournament and said it allowed him to focus his energy on preparing the team.
"A few years ago I would have read things and listened to things," added Southgate.
"Now I have to cut myself off, I can't put myself in that position because I lose focus.
"It allows me to be how I appear now with the team, they can see how we improve things and we are composed in what we are trying to do."
Southgate is expected to make at least one change from the Denmark draw with Conor Gallagher set to replace Trent Alexander-Arnold in midfield.
Luke Shaw joined in team training for the first time since arriving in Germany on Monday, but is not yet fit enough to feature at left-back.
