The Wildcats hit two out of every three shots they took from the field and shook off a slow start to put together their most solid showing of the season.
Goodwin and Poythress combined to hit 17-of-26 field goal attempts and Kentucky dominated the backboards 40-29.
Kentucky improved to 4-1 and kept the Blackbirds winless in their four outings. Nerlens Noel added 18 points for Kentucky and Julius Mays had 15.
Long Island had a three-point lead with 4 minutes left in the first half, but the Wildcats scored the final 15 points of the period and coasted from there.