Lucas Venuto, Brett Foy and Spenser Cobbold also scored for the Steelheads (7-4-1), who got 25 saves from Spencer Martin.
Daniil Zharkov replied for Belleville (7-4-1).
Tied 1-1 after the first, Mississauga scored three goals in the second period to open things up.
Bourne got his first in the OHL and Brace followed that up with a power-play goal. Venuto then also added his first career goal after stepping over the Belleville blue-line and ripping a shot past Charlie Graham.
Graham was replaced by Malcolm Subban to start the third after making 16 saves on 20 shots.
Subban gave up goals in the third period to Foy and Cobbold, finishing the night with 10 saves.