Brazilian striker de Oliveira Rafinha put Ulsan ahead after 51 minutes but Je Heon Song ensured the spoils were shared with his strike a minute from time.
The result means Ulsan are now nine and eight points behind leaders FC Seoul and Jeonbuk Motors in second place respectively.
Chunnam Dragons slipped to the bottom of the table after a thrilling 4-3 loss to Pohang Steelers.
Lucio da Silva Oliva Flavio opened the scoring with a sixth-minute penalty and put the Dragons ahead again in the 33rd minute after Jin Sung Hwang had levelled proceedings in the 15th minute.
But Pohang hit back after the interval and goals from Jin Yong Kim (49), Asamoah (51) and Sung Ho Park (75) put them 4-2 ahead and although Shung Hoon Jeong narrowed the deficit with an 81st-minute penalty, Chunnam could not find the equaliser.
Daejeon Citizen took advantage of Chunnam's defeat as their 1-1 draw with Gyeongnam moved them above the Dragons on goal difference.
Gyeongnam took the lead through Ii Rok Yun's 16th-minute strike but Tae Yeon Kim's 67th-minute penalty grabbed a point for the hosts.
Goals from Joon Jae Nam (21) and Joon Tae Park (62) gave Incheon United a 2-0 win over Gangwon FC.