World No.121 Chung Hyeon sealed the tie for the visitors after he beat Danai Udomchoke 6-4, 6-1, 6-1 to give South Korea the third and decisive point at the National Tennis Centre, Muang Thong Thani.
Warit Sombutnark won the dead rubber for Thailand after he beat Lim Yong-Kyu 6-4, 6-3.
The Koreans now advance to a second-round tie against Uzbekistan, who received a first-round bye, in July.
Thailand will meet the losers of that tie in the first round play-off in September. If the opponent is South Korea then Thailand will automatically be demoted to the second round play-off according to the rules.
Danai, who has been carrying an ankle injury, said it might be his last Davis Cup appearance.
"At first, we wanted Sonchat or Sanchai [Ratiwatana] to play but they had a tough doubles match on Saturday. So I decided to play and tried my best. Unfortunately my knee and ankle started to hurt.
"I think this is my last time playing Davis Cup for the country. I am not capable of playing the best-of-five format anymore because of the injuries I've had," he added.
Meanwhile world No.1 Novak Djokovic powered Serbia into the World Group quarter-finals on Saturday while France also advanced.
Djokovic's Serbia took an unbeatable 3-0 lead over Croatia, while France ensured a victory over Germany in first-round play.
But defending champions Switzerland, missing big guns Roger Federer and Stan Wawrinka, face being dumped out of the team tournament in the first round as Belgium took a 2-1 advantage by winning the doubles in Liege.
The United States kept their hopes alive against Great Britain in Glasgow despite Mike and Bob Bryan, the top doubles team in the world, being pegged back to two sets all after winning the opening two frames against Jamie Murray, the brother of Davis Cup teammate Andy Murray, and Dom Inglot.
The twins kept the Americans in the contest ahead of the reverse singles with a 6-3, 6-2, 3-6, 6-7 (8/10), 9-7 victory.
There were early celebrations in Kralijevo after Djokovic came in as a last-minute doubles replacement for Viktor Troicki and alongside Nenad Zimonjic convincingly beat Marin Draganja and Franko Skugor 6-3, 6-4, 6-1.
Brazil seized a 2-1 lead in Buenos Aires after Marcelo Melo and Bruno Soares defeated Diego Sebastian Schwartzman and Carlos Berlocq 7-5, 6-3, 6-4 in doubles.
In Frankfurt, France assured themselves of a quarter-final berth when Julien Benneteau and Nicolas Mahut posted a 6-4, 6-3, 6-2 doubles success over Benjamin Becker and Andre Begemann.
Gilles Simon and Gael Monfils had put France 2-0 up after beating Jan-Lennard Struff and Philipp Kohlschreiber respectively on Friday.
In Ostrava, the Czech Republic kept their hopes alive 2-1 with Jiri Vesely and Adam Pavlasek beating Australia's Lleyton Hewitt and Samuel Groth 1-6, 7-6 (7/2), 3-6, 7-6 (7/4), 6-2 in the doubles.
Thanasi Kokkinakis and Bernard Tomic had put the Australians 2-0 up on Friday. Bangkok Post/AFP