A deflected strike from Emil Forsberg was enough for the Swedes to advance to the last eight for the first time in 24 years.
Speaking to reporters after the game, and quoted by Reuters, Andersson said: "Now it’s about taking the next step. We’re not satisfied with this, and nothing more. We want to win the next match as well."
The 55-year-old issued a warning against complacency: "If you start to lower the bar, the level of ambition, that’s not going to make for satisfaction."
Andersson was full of praise for his side's team spirit after their third victory in four matches in Russia. "This team personifies the approach we all share - we work for each other on and off the pitch, and I’m incredibly happy that it’s paying off."
When asked whether Sweden's opponents had underestimated them, Andersson replied: "I think you’d have to ask everyone else that question. We know we are a good team, that we’ve earned our successes."
He added: "We’ve worked this way throughout, we’re continuing the same way, and what other teams and countries think about that is not terribly interesting."
Jubilant Swedish fans stayed behind in the stadium in St Petersburg and sang their manager's name long after the end of the match. Reacting to his fans' adulation, Andersson struck a modest note.
"It’s a surreal feeling, to be standing in St Petersburg on the pitch afterwards and people are calling out my name," he said, adding: "That feels strange - during the match I’m very much focused on the game."
Sweden will face England in their quarter-final on Saturday, after the Three Lions earned a dramatic penalty shootout win over Colombia on Tuesday evening.