Klinsmann was appointed last month to replace Paulo Bento and has not won in either of his first two games in charge having seen his side draw 2-2 with Colombia on Friday.
Defender Sebastian Coates gave the Uruguayans the lead in the 10th minute when he rose above the Korean defence to head in Federico Valverde's corner.
But Hwang In-beom levelled six minutes after the interval when, with the Uruguayan defence in disarray, Lee Ki-je cut the ball back for the midfielder to slot in.
Vecino restored his side's lead in the 63rd minute when he pounced on a loose ball in the six-yard box after Jo Hyeon-woo had parried Joaquin Piquerez's free kick from outside the area.
The Koreans thought they had equalised again when Kim Young-gwon rose above goalkeeper Santiago Mele to head in with 15 minutes remaining, only for referee Yusuke Araki to judge the defender had handled the ball after a lengthy VAR check.
Celtic striker Oh Hyeon-gyu then fell victim of a marginal offside call when he thought he had levelled seven minutes from time as the Uruguayans clung on for the win.
