Staples led 2-0 after one quarter but Weston tied it at 2-2 by halftime. The visitors were then outscored 6-2 in the second half.
Sean Fumai scored two goals for Weston. Troy Flynn and Richie Miller each netted one. Eli Page had an assist.
Grant Limone made 15 saves for Weston, which visits Newtown on Monday in the South-West Conference quarterfinals at 4 p.m.