The winners of the last eight tie in Dusseldorf on Saturday will face either Turkey or the Netherlands in the semi-finals.
England were hyped as pre-tournament favourites but have stumbled their way into the quarter-finals with a string of unconvincing performances.
Under Southgate they have never failed to reach the last eight in four major tournaments.
However, they are still aiming to end a 58-year wait for glory since lifting the World Cup in 1966.
"I would say that's a classic example of the sort of entitlement we have as a nation that creates drama and annoys our opponents," said Southgate when asked if England have to grasp a golden opportunity in Germany.
"Lots of nations who we might perceive as English people to be smaller have had far better records than us in terms of winning things and in terms of getting to the latter stages of finals."
England needed a stunning late equaliser from Jude Bellingham to avoid being embarrassed by Slovakia in the last 16 before winning 2-1 in extra-time.
That followed a disappointing group stage where Southgate's men were held by Denmark and Slovenia.
But the England boss is hopeful his under-par side will have turned a corner after their great escape in the previous round.
"There's been a lot of expectation on the team in the early part of the tournament especially, so I feel that the team, even in training, now look in a different place mentally. They look more fluid and I'm expecting us to play well tomorrow," added Southgate.
"The longer the players are here, the more belief they have, the less they're being affected by what's outside and they can see the opportunity."
Southgate is reportedly set to change formation with a shift to a back three in a bid to kickstart England's sluggish displays.
He will be forced into a least one change as Marc Guehi is suspended, while Luke Shaw could make his first appearance for club or country since February after being passed fit.
