The stand-in captain hit the winner 12 minutes from time as Belgium finished with a perfect three wins in Group H. They now play the United States in the last 16.
South Korean goalkeeper Kim Seung-Gyu could only parry a fierce shot by substitute Divock Origi and it went to the feet of Vertonghen, who buried the rebound and leapt into the crowd to celebrate.
South Korea had needed a big win to stand any chance of qualifying, and they were given hope when Steven Defour was sent off for a stamp just before half-time.
Belgian coach Marc Wilmots made seven changes from the team that beat Russia 1-0, giving Manchester United 19-year-old Adnan Januzaj his World Cup debut, to keep players fresh for the knock-out stages.
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South Korea showed urgency at the start and they had two hopeful penalty shouts waved away before Kim Young-Gwon, receiving the ball in the area from a free-kick, disappointingly blasted off-target.
Son Heung-Min received a great through-ball but was quickly closed down. Belgium showed little threat although Kevin Mirallas, bursting on to a pass from former Everton teammate Marouane Fellaini, had a goal chalked off for offside.
It looked like Belgium would take control but Ki Sung-Yueng brought a sharp save from Thibaut Courtois with a low drive before Defour cleared Koo Ja-Cheol's header off the line from the resulting corner.
The busy Mertens fired another chance well wide and then forced goalkeeper Kim Seung-Gyu to tip a free-kick over the bar after he surprised him by trying his luck from distance.