Mike Popoko had nine points and eight rebounds for the Seawolves (8-15, 3-8 America East), who snapped a seven-game road losing streak.
UMBC (9-15, 5-6), which has dropped two straight, got 14 points from Brian Hodges.
The Retrievers' total marked their second-fewest points in a home loss, the least since a 54-38 setback to Charleston Southern during the 1995-96 season.
Stony Brook scored 14 points off 15 UMBC turnovers, held a 32-8 edge in the paint and tallied 16 transition points while holding the Retrievers without a transition basket.
An early 9-0 burst helped Stony Brook to a quick 22-9 lead. Beauford had eight first-half points for the Seawolves, three baskets coming on transition lay-ups following UMBC turnovers.
The Retrievers trailed 22-13 at halftime, shooting 6-for-20 from the field for the opening 20 minutes and establishing a season low for points in a half.
Lucas scored 11 points as Stony Brook opened the second half on a 19-6 run for a 41-19 lead. He hit a pair of 3-pointers and capped the surge with a tip-in.