Howard, a freshman center, scored 16 points in the second half -- most of them near the basket -- and finished with 11 rebounds and three blocks for the Bulldogs (8-0).
Jon Goode had 21 points and seven assists for the Titans (4-4), who lost their third straight to Butler.
Betko's 3-pointer with 2:45 to play gave the Bulldogs the lead for good. Butler entered the game shooting 44 percent from 3-point range but finished 3-for-19 against Detroit.
Howard's tip-in on the possession after Betko's 3 forced Detroit to foul the rest of the way.
The Bulldogs led 20-17 at halftime despite shooting just 27 percent from the field.
Mike Green added nine points for Butler.