Trae Anderson scored a career-high 31 points for Eastern Illinois (2-5).
Burks made a layup to stretch Marshall's lead to 70-53 with eight minutes remaining. Casey Teson and A.J. Riley made back-to-back 3-pointers to spark a 17-5 surge to pull Eastern Illinois to 75-70 with two minutes left.
Anderson scored three layups, Marshall made only 5 of 10 free throws, and the Thundering Herd led 80-76 with 26 seconds to play.
Taylor split another pair of free throws, and Stevie Browning forced a turnover and made a free throw to seal it.