In a contest desperately short of quality in the final third, Forsberg -- Sweden's brightest player -- broke the deadlock with a low finish after 66 minutes, and Sweden thought they had won a stoppage-time penalty, only for referee Damir Skomina to award a free-kick from 18 yards following a VAR-review.
The only goal came when Forsberg played a one-two with Ola Toivonen, sold Arsenal's Granit Xhaka a dummy and rifled home a low finish via the boot of Switzerland's Manuel Akani to send the Swedish fans behind the goal delirious.
The RB Leipzig winger, who has been described as the "anti-Zlatan" because he is so humble, then denied Akani an equaliser, clearing the centre-half's header off the line from the Swiss corner. The loser's best chance of the game came in stoppage-time, when substitute Haris Seferovic worked Sweden goalkeeper Robin Olsen with a low header.
Before the match, Swedish captain Andreas Granvquist admitted they were without "one of the best footballers in the world" in Zlatan Ibrahimovic but this victory ensured Sweden have gone further than at any World Cup since USA '94, when they reached the semi-final. They will have a chance to match that achievement against either England or Colombia, who meet in the Russian capital later tonight, in the quarter-final in Samara on Saturday.
Sweden might have led inside ten minutes, only for Albin Ekdal to blaze over the bar after Marcus Berg's shot was blocked, and midfielder Ekdal was guilty of another glaring miss minutes later, volleying over with the goal gaping. Berg also forced Swiss goalkeeper Yann Sommer into a fine one-handed save, as Sweden edged the first half.
Switzerland were also guilty of wayward finishing and Blerim Dzemaili shot high and wide from a good position. Swiss chances were few and far between after the break, however, and Forsberg's goal left them chasing the game.
As they pushed for a stoppage-time leveller, Sweden sub Martin Olsson raced forward with only the goalkeeper to beat, only to be nudged in the back by Michael Lang. Referee Skomina originally pointed to the spot but overruled the decision after consulting the VAR, and Sommer's save from Toivonen's resulting free-kick was the final act of the match.
In an act of karma, Mikael Lustig, who had vowed before the match to get Xhaka sent off, was booked in the first half for needles foul on Josip Drmic and will miss the quarter-final.