After sloppy but effective wins over Russia and Algeria in their first two matches, the Belgians entered their group-stage finale needing only a draw to win the group.
Vertonghen netted the only goal of the contest 13 minutes from time and the Belgian defense held out for the clean sheet and the victory, which sees Marc Wilmots' team join Colombia, Argentina and the Netherlands as the only teams to take nine points from their group stage games.
Belgium will face the United States in the knockout round in Salvador next Tuesday, while South Korea finishes on the bottom of Group H with just one point.
Fielding a starting 11 which consisted mostly of bench players, Belgium had the better of the play early and Napoli man Dries Mertens had the first golden opportunity to get on the score sheet 25 minutes in when the ball fell kindly to his feet, but his strike sailed well over the crossbar.
South Korea responded with a chance on the half-hour mark as Ki Seung-Yeung had a go from well beyond the penalty area that forced Belgium goalkeeper Thibaut Courtois to make a nice diving stop.
Belgium went a man down on the stroke of halftime after Steven Defour was shown a straight red card by referee Ben Williams for a clear stomp on the leg of Korean forward Kim Shin-Wook.
Wilmots' men finally took the lead 13 minutes from time when Divock Origi ripped a strike on target that couldn't be handled by Korean goalkeeper Kim Seung-Gyu. The rebound fell nicely to the on-rushing Vertonghen, who fired it home with ease.
South Korea created little in the way of scoring chances over the final 13 minutes and bowed out of the World Cup, while Belgium will get ready to face the United States next week.