Victory would have taken Jeonbuk level on points with leaders FC Seoul, with a superior goal difference, with one game remaining before the new league split.
But they fell behind to Han Kyo-won's 55th-minute goal and though Jin Kyung-sun equalised, Nam Joon-jae grabbed Incheon's winner.
Third-placed Suwon Bluewings needed a last-minute equaliser to earn a 2-2 draw at home to lowly Daejeon Citizen.
Seo Jung-jin quickly cancelled out Kevin Oris' opener, but Kim Hyeung-bum restored Daejeon's lead in the 70th minute.
Ha Tae-goon struck in the 90th minute but it could only earn a point for his side, which left them nine off the summit and lifted the visitors three points above the foot of the table.
Late goals from Everton Santos and Jael Ferreira earned Seongnam Chunma a 2-1 win away to Jeju United, who had taken a first-half lead through Song Jin-Hyung.