Choi Yong-soo's charges are now unbeaten in nine games in all competitions and leapfrogged Ulsan in the table on Wednesday courtesy of Koh Myong-jin's last-gasp strike.
Hwang Ui-jo had put the home side in front eight minutes into the second half at the Tancheon Sports Complex, but Ko Kwang-min levelled 13 minutes later.
Seoul have secured victories over Pohang Steelers and Jeonbuk Motors - the top two in the K-League - courtesy of late goals and they were at it again, as Koh snatched all three points to move the visitors into the top half of the table.
Choi's men are riding on the crest of a wave, as they also have a two-legged AFC Champions League semi-final against Western Sydney to look forward to while they also flex their muscles in the K-League.
Leaders Jeonbuk dropped points again as they were held to a 1-1 draw at second-bottom Busan I'Park.
Lee Dong-gook put the visitors in front 58 minutes in, but Fagner's penalty seven minutes from time salvaged a point for Busan - who had Kim Ik-yun sent off for a second bookable offence in stoppage-time.
The second-placed Steelers are now only a point behind the leaders after they responded to their defeat against Seoul by winning 1-0 at Jeonnam Dragons.
Moon Chang-jin was the match-winner, scoring the only goal of the game just eight minutes in and the home side were unable to respond at Gwangyang Stadium so they drop a place to fifth in the table.
Jeju United are up to fourth as a result of their 2-1 win at Sangju Sangmu, while third-placed Suwon Bluewings beat Ulsan 2-0 and Incheon United were held to a goalless draw at bottom side Gyeongnam.
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