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.
But they failed to create clear-cut opportunities against an obdurate Belgian defence and depart the tournament winless, as did regional rivals Japan, Iran and Australia.
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.
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.
A Mexican wave after19 minutes indicated the crowd’s thoughts were elsewhere, but Dries Mertens regained their attention when he found Mirallas in the box and then miscued wildly from close range.
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.
Defour got a straight red card from Australian referee Ben Williams for a two footed challenge on Kim Shin-Wook. The Porto midfielder made no protest and walked straight off.
With Russia leading Algeria 1-0, South Korea needed at least two goals to go through and coach Hong Myung-Bo brought on striker Lee Keun-Ho at half-time as they went for broke.
Ki’s sensational cross-field pass picked out Lee Chung-Yong on right and substitute Lee then headed over as South Korea threatened.
Bayer Leverkusen’s Son ran into space down the left and found Lee Chung-Yong, whose shot was blocked, and then from the right, he floated a cross-cum-shot onto the Belgian bar.
Belgium were doing most of the defending but Manchester United’s Fellaini had a strong penalty appeal denied when he was checked by Hong Jeong-Ho.
It felt like a goal was on its way and when Ki rose to meet Son’s corner from the right, Korean fans held their breath. But he mistimed his header which was easily gathered by Courtois.
Januzaj was withdrawn on 59 minutes and replaced by Origi, Belgium’s scorer in the 1-0 win over Russia. Shortly afterwards, Fellaini should have broken the deadlock but he put a simple header wide.
Despite their extra man, South Korea could not penetrate the packed Belgian defence and Lee Chung-Yong and Koo both saw dangerous opportunities snuffed out before they could shoot.
And they paid for their failures when Vertonghen scored.