Paul Baxter's 3-pointer with 8:20 left in the game capped a 20-7 run and extended the Bearkats' (22-9) lead to 57-41.
The Golden Eagles (17-16) answered with a 10-3 stretch to close within nine points with 2:59 left. But Peters and Ransom combined to make 9 of 10 free throws in the final 1:18, advancing Sam Houston to a semifinal matchup with No. 2 seed Texas A&M-Corpus Christi (17-14) on Friday.
Baxter was 3 of 3 from 3-point range while Ransom and Peters were each 3 of 5 from distance.
Korey Billbury had 20 points and seven rebounds to lead Oral Roberts, which returns to the Summit League in 2014-15 after a two-year run in the Southland Conference.