With 10 minutes to go, Embolo had a header scrambled away from nearly on the line by Forsberg who, along with Lustig, was substituted in a double change as time ticked away, Emil Krafth and Martin Olsson coming on.
Switzerland last made it to the quarter-finals of the World Cup in 1954 when the tournament was held on home soil.
Sweden may have won on the day, but they will need some serious improvement if they are to progress to the semi-finals.
They will now await the victor of Colombia or England and, having seen off the Netherlands and Italy in qualifying, and then finished above Germany in Group F, Andersson says there is still more to come. Their next opponents would be foolish to take them lightly.
The South Americans had been getting away with blatant holding and wrestling at each of England's many corners and the referee's patience finally snapped when Carlos Sanchez hauled down tournament leading scorer Kane once too often.
In the first half, the team failed to convert their chances, although two caused 14 strikes on goal, but after the break, the fans saw a goal which became victorious.
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When Rodrigo charged down the right flank, he forced a good save from Igor Akinfeev , before Dani Carvajal mishit the rebound. Russian Federation will now meet Croatia in the Quarter Final at the Fisht Olympic Stadium, Sochi on Saturday, July 7, 2018.
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One cross was expertly delivered in to the area only for Steven Zuber to head over while another passed dangerously across the front of goal and wide.
Sweden have qualified for their first World Cup quarter-final since 1994.
Emil Forsberg's deflected second-half goal gave Sweden a 1-0 victory over Switzerland to book their place in the World Cup quarter-finals for the first time in 24 years.
"It's my job to decide on the players who need to perform, in the case of a penalty shootout situation". "What other teams and countries think about that, you'll have to ask them". "We had today good chances to go through and we didn't take the chance". "We make sure we work hard for each other on and off the pitch".
"We have a gameplan and they do not stray from it one millimeter", said Andersson of his team, aside making up for its lack of superstars - notably Zlatan Ibrahimovic, who retired from worldwide football after Euro 2016 - with unity and spirit.
"We have a team where the chemistry is right, that really goes together, and this is a guarantee for success". But we need to digest this quickly as we have another game coming up soon, on Saturday.
Both teams were wasteful in possession and guilty of the sort of poor finishing and unimaginative midfield play that had boos and whistles ringing around the St Petersburg stadium from as early as the 25th minute. Switzerland will be without right-back Stephan Lichtsteiner and central defender Fabian Schar while Sweden midfielder Sebastian Larsson is also out after picking up two yellow cards in the competition's first phase. Although Janne Andersson's team are a collection of functional individuals who operate together like clockwork, Forsberg is the big name and in Saint Petersburg, the RB Leipzig midfielder nearly didn't score only to be helped by Swiss defender Manuel Akanji's deflection.