Guard Liam Aston led the scoring with 18 points but Benoit Hayman (16 points), Josh Petermann (15) and Joe Cook-Green (14) all made major contributions.
Centre Sam Timmins had a quiet opening half but finished strongly, grabbing 15 rebounds and chipping in with nine points.
His opposite, Yuat Alok, dominated the early stages.
The Breakers development player amassed 19 rebounds and 14 points, while Tall Black rookie Tai Wynyard also registered a double-double with 13 rebounds and 12 points.
Rangitoto owned the opening quarter, shooting out to a 25-16 lead.
But OBHS responded with a superb defensive effort in the second half, holding its opponents to just 19 points.
Assistant coach Brent Matehaere said his side was left reeling after the opening period but grew in confidence during the second spell before seizing the initiative in the third quarter, outscoring Rangitoto 20-6.
''The defence just told,'' he said.
''Sam made about three blocks in a row ... and that was the moment in the game where the boys all of a sudden realised they were in a ball game.''
OBHS will play the winner of the late semifinal between Auckland's Westlake Boys' and Wellington's St Patrick's College.
