VMI (3-5) led by as many as six points in the second half, and held a 63-59 lead with about nine minutes left.
Army (5-2) responded with an 8-0 run to turn a four-point deficit into a four-point advantage to take the lead for good.
Army was 8 of 8 at the free throw line in the final two minutes.
The teams traded the lead back and forth in the first half, heading to the break tied 37-all.
Brown shot 8 of 17 from the floor. Walker converted 8 of 11 field goals and 7 of 10 free throws.
Army outrebounded VMI 45-31.
Maryland point guard Melo Trimble has withdrawn from the NBA draft and will return to College Park for his junior season, sources told ESPN.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.