The 1966 winners will face either Russian Federation or Croatia on July 11 at the Luzhniki Stadium for a place in the final, and Southgate has his eyes on the home nation.
Headers from Harry Maguire and Dele Alli fired an impressive England into the World Cup semi-finals for the first time in 28 years after they eased their way to a 2-0 victory over a dogged but disappointing Sweden on Saturday.
The winning goal from Maguire of Leicester City arrived just on 30 minutes, and it was another textbook set piece for England. He sent in the feed, finding Maguire straightaway from the goal. England had not kept a clean sheet in their four previous matches and had Pickford to thank for ensuring they did not concede this time. Raheem Sterling may come under criticism for freezing in front of goal twice in as many minutes, but his driving runs and pace were key to sustaining the pressure on Sweden's backline.
Sweden finally showed some bite with Marcus Berg forcing Jordan Pickford to dive to his right and keep out a bouncing header in the second minute of the final half.
Early on, the Swedes, contended with England enjoying possession, used their numbers in the midfield to slow down play, thus nullifying the threat of pace that England possesses.
Confidence, optimism and expectation is sweeping the country as England supporters dare to dream football may be on its way home.
"The resilience of the team and togetherness of the team today was crucial".
Concerned England fans make the same joke about the referee against Sweden
This was England's most complete performance of Russian Federation 2018 in what was a highly-credible team effort. Former captain Shearer earlier lauded England's "professional" and "thorough" performance against Sweden .
England manager Gareth Southgate said it was an "incredible feeling" to be through to the semi-finals.
"I feel privileged to be in this position", he said.
The Sweden team got in a huddle after the game and Andersson spoke to the players.
Among those reacting to the historic win on Twitter was Prince William, who posted to the Kensington Palace account. This truly is a team making its own history.
British Airways also tweeted, posting a picture of a mocked-up boarding pass showing football "coming home".
The overachieving hosts, the lowest ranked nation in the tournament, were trying to make it to the World Cup semifinals for the first time since the Soviet Union team finished fourth at the 1966 tournament in England.
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