CU took command of the game with a 21-6 run midway through the second half, a stretch in which Roby nailed two treys.
Fellow freshman Marcus King-Stockton turned in his second strong effort in as many games, subbing for an injured Julius Ashby and playing a career-high 18 minutes. King-Stockton shot 4-of-4 from the field and ended with 11 points and a game-high eight boards.
Jayson Obazuaye added 16 points for the Buffs, while Marcus Hall had six assists. Andy Osborn played six minutes and scored two points, his first action since injuring his ankle earlier in the season.
Mike English and Quinton Day led UMKC with 18 points each.
CU moved to 5-2 with the win, while UMKC fell to 0-6 on the season. The Buffs host Radford Thursday (7 p.m.).