Veteran Swedish skipper Granqvist scored the only goal of the game from the spot after Viktor Claesson was upended by Korea Republic substitute Kim Min-woo Kim in the 62nd minute.
Despite passionate appeals, El Salvadorean referee Joel Aguilar initially waved away Swedish penalty claims.
But after consulting the video, he pointed to the spot.
It was the third VAR penalty awarded in Russia 2018 after France and Peru also benefited from video referrals.
Sweden never really convinced that they would score prior to the penalty despite their domination and Korea Republic did not manage one shot on target.
However they should have equalised in the 90th minute but Hwang Hee-chan put a free header wide, spurning the Koreans' best chance of the game by far.
The victory leaves Sweden joint top of the group after Mexico's surprise victory against defending champions Germany on Sunday.
Next up for the Swedes is an intriguing encounter against Germany in Sochi on June 23.
Korea Republic started the game brightly, forcing the first corner in the fourth minute, but were gradually forced back by an attritional Swedish side.
It was the Europeans who came closest to breaking the deadlock when forward Marcus Berg was denied by a fine point-blank save Jo Hyeon-woo in the 20th minute.
Berg was denied again nine minutes later by a desperate last-ditch block from Kim Young-gwon.
Despite their increasing dominance, Sweden also demonstrated why they have struggled to score goals.
Berg, again, and Claesson went close before the end of the half, but it was clear for everyone to see why they had only scored once in their four matches prior to Russia.
So, Group F after one round of fixtures...
1) #SWE
2) #MEX
3) #GER
4) #KOR #WorldCup pic.twitter.com/GNICy1p5bo— FIFA World Cup (@FIFAWorldCup) June 18, 2018
Korea Republic threatened only on the break where Son Heung-min's pace troubled 33-year-old Granqvist, who was without defending partner Victor Lindelof, who was ill.
Resilient Sweden held firm after the goal and the Hwang late scare aside, and a half-hearted Korea Republic appeal for a penalty, their defence rarely looked troubled.
The huge celebrations, the first match played in the $290 million stadium, at the end showed the importance the Swedes attached to this match.
Before the game, Korea Republic's coach Shin Tae-yong called it a "must win" game but the defeat leaves his side contemplating an early exit.
Sources: AFP, FIFA via Getty Images
Source: the-afc.com