Emil Forsberg's deflected shot proved the difference at the Saint Petersburg Stadium, where Sweden shaded a scrappy contest to reach the last eight for the first time since 1994.
They will now await England in the quarter-finals, and Andersson says there is still more to come.
“We're not satisfied with this,” he said.
“We want to win the next match as well. If you start to lower the bar or your ambition that's not going to be satisfying.”
“Hopefully in the future I will look back on this with pride but for now, I focus on the next match on Saturday.”
Forsberg has been tasked with taking over the creative mantle since Zlatan Ibrahimovic retired from international football two years ago and the 26-year-old delivered when it mattered.
But Sweden's success has been a collective effort, with their organization, determination and spirit so far coming up trumps in Russia.
“We really are a team as a whole,” Andersson said.
“The team fighting for the team is what prevails for the people out on the pitch, and all of us on the sidelines.”
Switzerland, meanwhile, exit the World Cup in the last 16 for the third time in four attempts and their coach Vladimir Petkovic admitted his team were well below their best.
“There was something missing in that match,” Petkovic said.
“We've seen with the Swedish team that their opponents have had a hard time finding those emotions against them.”
“It was a difficult task and after conceding we just couldn't turn the match around.”
Meanwhile at Moscow's Spartak Stadium, England broke its penalty shootout curse (it had lost six out of seven times when taken to shootouts at major tournaments), beating Colombia.
They must now re-set ahead of Saturday's match in Samara against Sweden, who should not be underestimated.
The Netherlands and Italy both fell to the Swedes in qualifying while Mexico and Germany each finished beneath them in Group F.
England, it should be remembered, were also far from perfect. There was a nervousness in the last 10 minutes when they suddenly dropped deeper and their passes became more frantic.
But for all that, it cannot be ignored what this win will do for England's momentum and morale.
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