Paul Kerr scored the opening two tries before James McConnell added a third as MCB went in 22-9 ahead at half-time.
Johnny Jordan, Thomas Gallagher and Kerr touched touch for MCB in the second half while BRA tries came from Bradley Luney and Matthew Dalton.
MCB will take on holders RBAI in the St Patrick's Day final.
RBAI brushed aside the challenge of Ballymena Academy in the first semi-final at the same venue on Monday.
Niall Armstrong went over for three of their seven tries in a one-sided 41-8 win.
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Methodist College Belfast cruised into the Schools' Cup decider thanks to a six-try victory over Belfast Royal Academy at Kingspan Stadium on Tuesday.