Adilson gave Seoul a 29th-minute lead but Seok Lee quickly equalised and Diogo Acosta put Incheon ahead six minutes after half-time.
Park See-heong levelled the match once more but Moon Sang-yun grabbed the winner in the 80th minute.
Jeonbuk Motors and Suwon Bluewings both made it back-to-back wins to open their respective campaigns.
Jeonbuk beat Ulsan Hyundai 2-1, Park Hee-do with the winner after Leonardo's opener was equalised by Han Sang-woon.
A solitary goal from Kim Do-heon in the 11th minute was enough for Suwon to beat Gangwon FC.
Cho Chan-ho scored twice, adding to Ko Mu-yeol's opener, as Pohang Steelers beat Daejeon Citizen 3-0.
Jeju United came from behind to draw 1-1 with Seongnam Chunma, Park Jin-po's own goal cancelling out Yun Young-sun's opener.