The Football Association says it won’t sell tickets for the match in Rome due to Covid restrictions.
There were jubilant scenes as England fans celebrated the historic win.
Southgate said the team started talking about Saturday in the dressing room.
“It’s a dangerous moment for us,” he said.
“We’ll have that warmth of success and the feeling around the country that we’ve only got to turn up to win the thing – and we know it’s going to be an immense challenge from here on.”
England’s win was their first in a knockout tie over Germany since the 1966 World Cup final.
And there was a carnival atmosphere across the nation after two late goals by Manchester City forward Raheem Sterling and Tottenham striker Harry Kane secured their victory.
Former England captain Alan Shearer said he couldn’t remember Wembley ever being that loud for an England game, despite the reduced capacity, describing the atmosphere as “incredible”.
He told BBC Breakfast: “We asked before the tournament that this squad and this manager, if they could just give the whole nation something to smile about and to feel good about because we’ve all been locked up for the last 18 months… that was well worth the wait last night, it was a great occasion.”
If England beat Ukraine, they will play the Czech Republic or Denmark in the semi-finals. The tournament draw also means that if England get to the final, Saturday’s trip to Rome will be the only one of their seven matches not at Wembley.
Shearer added: “I’m trying not to get carried away like the rest of the nation but the draw has fallen well for us. If we could have handpicked a draw then it would have been this.”
An animated Duke of Cambridge was among the roaring supporters inside Wembley – the FA president throwing his arms aloft alongside his seven-year-old son, Prince George, and the Duchess of Cambridge, as each goal went in.
They were joined in the Royal Box by stars including singers Ed Sheeran and Ellie Goulding and former England captain David Beckham.
Also among the crowd were David Baddiel and Frank Skinner, who famously sang England anthem Three Lions with its familiar refrain, “Football’s coming home”. Baddiel said it was “beautiful” to hear the crowd belting out the song at the end.
He told Radio 4’s Today programme Tuesday’s match was “so tense for so long” but after seeing Sterling score the opening goal everyone around them went wild and they were “blind with joy for a little while”.
There were also jubilant scenes elsewhere as fans kitted out in England football strips and costumes gathered in designated fan zones, pubs and bars to see the historic win.
A party atmosphere broke out in Trafalgar Square in London after England scored their second goal, with several men moved to rip off their shirts and start dancing on tables.
England fan Matt Champion from Bristol – who was celebrating his 32nd birthday – said the result was “unbelievable”.
“I’m very, very happy. This is the best birthday party I could have ever organised.”
Fellow supporter Jay Gittelmon, 20, from north-west London, said of Sterling’s opening goal: “I’ve never felt anything like that, it’s the best moment of my life.
“I had to use my inhaler as I ran out of breath.”
Later, fans took to the streets of London to celebrate, with crowds gathering in Leicester Square before dozens of supporters climbed up on to the Shaftesbury Fountain in Piccadilly Circus.
England’s game on Saturday will be the first they have played at a ground other than Wembley since the Euros started.
They have won three of their four games so far, while drawing with Scotland, and have not conceded a goal in the tournament. It is the first time a team has gone through their first four games of a European Championship without letting their opponents score.
Germany join world champions France and Euro 2016 winners Portugal as some of the heavyweights exiting the tournament.
Can I go to Rome for the quarter-final?
England fans hoping to jump on a plane to Rome to see their team’s showdown with Ukraine are likely to be disappointed due to current coronavirus restrictions.
Passengers will need proof of a negative coronavirus test taken in the 48 hours before travelling.
Then, upon arrival, UK passengers must fill in a European Union passenger locator form before isolating for five days.
This means that even if you were to touch down first thing on Wednesday, the earliest you would be out is Sunday – subject to a second Covid-19 test – meaning you will probably be watching the game from a hotel room.
The Football Association has said it will therefore not be selling tickets for the match in Rome. It will try to get tickets to England fans who are already in Italy via the British Embassy.
The only people able to avoid having to isolate are members of transport crews and those who can prove they are in Italy for “work, health or emergency”.
If you fall into those categories you can enter Italy but must leave within five days.
Italy is on the UK’s amber travel list, meaning travellers must self-isolate for 10 days upon return and book Covid-19 tests for day two and day eight of their quarantine before setting off.
Passengers can also use the “test to release” scheme where they pay for a private Covid-19 swab on day five of their quarantine – if the results on both day two and day five are negative, they can end quarantine early.
The government website says people should not travel to amber list countries or territories.
After a difficult 15 months because of the pandemic, England defender Harry Maguire said he was delighted to see happy faces on football fans again.
“We put a smile on a lot of people’s faces, that’s what we do, that’s why we play football,” he said.
“But come Saturday we know it is business again and we can take the momentum and confidence from the deserved victory we have managed to get.”
If England can overcome Ukraine then they will face either the Czech Republic or Denmark for a place in the final at Wembley.