Nerlens Noel contributed a double-double of 11 points and 10 rebounds as the Wildcats (8-3) racked up their fourth straight win. Goodwin also grabbed six rebounds.
Two straight baskets by Harrow early in the second half triggered a 25-4 run that iced the game for Kentucky.
Alex Poythress chipped in nine points and nine rebounds for the Wildcats.
Elijah Pittman struck for 20 points and Nigel Spikes hauled down 16 rebounds for the Thundering Herd (7-6).