The first meeting between the nations at a major tournament simmered for 65 minutes in St Petersburg until Forsberg's fortunate strike handed Sweden a crucial lead.
Forsberg's powerful but seemingly unthreatening strike from the edge of the box was deflected off defender Manuel Akanji's left foot to leave goalkeeper Yann Sommer helpless.
The Swiss finished the game with 10 men after Michael Lang was sent off for pulling down Martin Olsson as he raced through on goal.
Sweden have reached the knockout stage in each of their last four World Cup appearances (in 1994, 2002, 2006 and 2018), athough they've only reached the quarter-finals once since 1958.
Janne Andersson's team were slightly the more aggressive and created the better chances.
The Swiss had suffered just one defeat in their last 25 matches - a defeat to my beloved Portugal in the qualifiers, which meant that the Portuguese qualified for the tournament automatically while Switzerland had to progress via a play-off against Northern Ireland, which was won through a controversial penalty.
There can be no doubting Switzerland are good at what they do when playing to their strengths; they will continue to grow under the stewardship of Petkovic.
While that might offend the purists, Sweden midfielder Albin Ekdal has already responded to criticism that his side are not the most easy on the eye. The Swedish striker should have done so much better with that opportunity.More news: Trump Narrows SCOTUS Choices To Three
More news: A new in-game message confirms Fortnite's rocket will launch tomorrow
More news: India look to kick off English summer on winning note
"It brings tears to my eyes", Forsberg said, "and makes me so proud".
Granit Xhaka tried his luck from distance for Switzerland but failed to hit the target before Dzemaili put another attempt over after combining nicely with Steven Zuber. Marcus Berg found himself in a great position after being played in by an Ola Toivonen flick but he sliced wide.
Switzerland looked to be more intent on seizing that opportunity at the outset, controlling possession in the first half and with much of the play flowing through Xherdan Shaqiri in midfield.
A small detail from a terrible match but indicative of its low quality and indicative of what England will face in the quarter-final of the World Cup this weekend.
And even when substitute Haris Seferovic did guide a header towards goal he was denied by a good save from Robin Olsen. With 12 minutes remaining, Francois Moubandje's header was blocked on the line by Forsberg and hacked clear by Andreas Granqvist, his captain.
Switzerland's Ricardo Rodriguez also missed a sitter in the 32nd minute from a free-kick but Olsen, the Sweden keeper held on.
"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".
All in all, the match was very far from being a classic, but the Swedes won't care about that.