After a back-and-fourth first half, Devon Atkinson's jumper drew Morehead State within 54-52 with 4:03 remaining in the second half. But Scarver hit his lone 3-pointer and Kane converted a fast-break layup to put Marshall up 59-52.
Morehead State (4-3) attempted to rally but Marshall hit 11 free throws over the final 1:17 to put the game away. Scarver had 10 of those free throws while Tinnon had the other.
Marshall has won 12 of the last 14 meetings, including the last 10 at home, against Morehead State. The Thundering Herd shot just 33.3 percent (20 of 60) but outrebounded Morehead State 53-46.
Jarrett Stokes led Morehead State 14 points.