A tight last 16 match was opened up by Forsberg in the 66th minute, when he found some space on the edge of the area and fired in a low drive which Yann Sommer would have fielded, but Manuel Akanji deflected it into the other side of the goal.
There was talk before the match that there must be something about Vladimir Petkovic's half-time team talks that injects fresh energy and impetus into his side, as they only really seem to get going after the first 45 minutes have concluded - four out of Switzerland's five World Cup goals have come in the second-half.
With the result, Sweden booked their place in the last eight of the World Cup for the first time since 1994, when they went on to finish third.
There had been little between the sides throughout the 90 minutes.
At the other end, Blerim Dzemaili wasted Switzerland's best chance when he fired over from 12 metres, while Xherdan Shaqiri had an bad day, sending long balls from the wing sailing over the heads of his team mates with alarming regularity.
Despite deploying two strikers in Ola Toivonen and Marcus Berg, the system works by sitting off and stifling the opposition with a hardwired midfield and defensive line that looks to hit the two front men whenever possible.
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Albin Ekdal managed to outdo his team-mate with an even worse miss as he blazed a chance over from close range just before half-time. He had another opportunity before the break when he was picked out at the far post by Mikael Lustig.
Shaqiri tried to turn provider after 15 minutes, but his cross from the right was over the leaping Valon Behrami and another centre was headed over by Steven Zuber.
Victor Lindelof slipped on the ball in the first 10 seconds and that set the tone for the first half, which was largely a collection of mishit passes, heavy touches and skewed shots. We'll never know if Sommer would have blocked Forsberg's shot had Akanji not attempted to deflect it, but he did appear to be in a good position.
That prompted Switzerland to show more urgency and Shaqiri had an effort deflected wide.
Sweden did not get the penalty they initially thought they had but it did not matter as time ran out for the Swiss.