Weston, the 15th seed, led 3-1 after one quarter and held a narrow 6-5 advantage at halftime.
Sean Fumai and Austin Drimal led Weston with three goals apiece. The former also had an assist.
Troy Flynn and Eli Page each scored two with an assist. Richie Miller scored one.
Russ Kirchenbaum, Grey Bradshaw-Mack and Spencer Turkel had one assist apiece
Grant Limone made 12 saves in goal for Weston, which will play seventh-seeded East Catholic in Manchester on Saturday at 3 p.m.