Rayshaun McGrew had nine points and 12 rebounds for Stony Brook (16-10, 7-4 America East), which made 16 of 26 (61.5 percent) second-half field goals.
A 3-pointer by Roland Nyama opened a 22-7 run which gave Stony Brook a 55-42 lead with 10:31 to play. Warney scored 10 during the stretch.
UMBC (3-21, 1-10) scored six straight to pull within 65-57 with 4:32 remaining, but the Retrievers make just 2 of 9 field goal attempts down the stretch and Puriefoy scored Stony Brook's final eight points to seal it.
Stony Brook outrebounded UMBC 41-20 and shot 55 percent overall.