Kobe, who had been the only team in the eastern league phase to start the competition with two consecutive wins, had been forced to fight back from three goals down to pull level, only for Kim to pounce at the death.
Lee Sang-Heon put the home side in front with an emphatic seventh-minute finish, and Mo Jae-Hyeon doubled the lead 14 minutes later with a powerful downward header following Kim Dae-Won's corner.
Song Jun-Seok completed a disastrous first half for Kobe with a little over two minutes remaining until the interval when he slid his shot through a crowd of players to beat Shuichi Gonda in the Kobe goal.
Kobe's introduction of Taisei Miyashiro after the interval swung the game back in their favour, the Japan international pulling one back within three minutes of the restart.
Two minutes later, Jean Patrick headed in Kobe's second, and Miyashiro was on target again with two minutes left, thumping the ball past Park Chung-hyo to pull his side level.
Kobe looked set to claim a point that would see them retain top spot in the standings, but Kim Gun-hee pounced in the 95th minute to secure the points for Gangwon.
Shanghai Shenhua, meanwhile, earned their first victory in the current competition as second-half goals from Luis Asue and Andre Luis sealed a 2-0 win over FC Seoul.
Asue headed in the opener 12 minutes after the interval, following Joao Teixeira's cross from the right, and Luis hit the winner a minute from time, rounding off a Shenhua counter-attack.
The results leave Ulsan HD on top of the eastern league standings after three rounds, the South Koreans holding a one-point advantage over Kobe and Gangwon following their 1-0 win against Sanfrecce Hiroshima on Tuesday.
Clubs finishing in the top eight positions in the 12-team eastern league will progress to the knockout rounds, which will be held in March. The quarter-finals, semi-finals and final will be played in a centralised venue in Saudi Arabia in April.