The first meeting between the nations at a major tournament simmered in the St Petersburg sunshine until Forsberg's fortunate strike handed Sweden a crucial lead.
The RB Leipzig playmaker's tame 66th-minute effort was heading straight at Yann Sommer until Switzerland defender Manuel Akanji stuck out a foot, sending the ball looping beyond his wrong-footed goalkeeper.
Forsberg also produced a well-timed clearance in his own penalty area to preserve the advantage against a Switzerland side who finished with 10 men, defender Michael Lang seeing red after clipping the heels of Martin Olsson just outside the penalty area when the substitute was clean through on goal.
Referee Damir Skomina initially gave a penalty deep into stoppage time before VAR intervened, though the change of decision made no difference to the final outcome of the contest.
The only disappointment for Sweden was a first-half booking picked up by Mikael Lustig, ruling the defender out of a last-eight clash against either Colombia or England in Samara on Saturday.
In contrast, Switzerland's run in Russia came to a rather subdued end. They had to change half of their defence for the knockout fixture, Stephan Lichtsteiner and Fabian Schar were both ruled out through suspension, yet it was in attack where they struggled the most.
— FIFA World Cup (@FIFAWorldCup) July 3, 2018
The reshuffled back four stood firm in the first half, though they relied upon goalkeeper Sommer's reactions to turn away Marcus Berg's first-time strike with his left foot.
They were also fortunate to see an unmarked Albin Ekdal fail to hit the target prior to the break, the Hamburg midfielder going for an ambitious volley instead of throwing his head at Lustig's inviting cross from the right.
Blerim Dzemaili blazed over at the other end to waste a rare sight of goal for Switzerland, who had developed a reputation as second-half specialists at this year's tournament.
Yet Vladimir Petkovic's side - who had scored four of their five goals in Russia after the interval - continued to struggle, only waking from their slumber after falling behind.
Ola Toivonen wastefully fired over when well positioned inside the penalty area before Forsberg finally broke the deadlock, albeit more through luck than judgement, his attempt clipping the unfortunate Akanji to beat Sommer.
2 - Sweden have won back-to-back World Cup matches for the first time since 1958, when they won the quarter-final and semi-final on their way to the final as host nation that year. Rising. #SWESUI #SWE #WorldCup pic.twitter.com/tAWajEqBDE
— OptaJoe (@OptaJoe) July 3, 2018
Switzerland responded to the setback by sending on forwards Breel Embolo and Haris Seferovic, and the latter at least tested Sweden goalkeeper Robin Olsen with a header in the closing stages.
Sweden were denied the chance to double their lead from the spot following Lang's late dismissal for taking out Olsson, with Sommer beating away Toivonen's driven free-kick after a lengthy delay - but one goal was enough for Janne Andersson's side to progress.
Photo: Francois Nel/Getty Images. Caption: Emil Forsberg celebrates scoring for Sweden against Switzerland