Alarm goes off: "To go off" means "to trip, to start sounding". Something has triggered the alarm, and it went off (started sounding, flashing lights, what not). This is about the ACTION that happens when someone trips the alarm. The alarm goes on - usually it means the alarm itself.

Is it alarm went off or on?

So 'the alarm went off' usually means the sound started, but 'turning the alarm off' means you stopped it. Saying 'the alarm went on' is not a thing, but 'turn the alarm on' means you set it to 'go off' at a later time.

Why do we say the alarm went off not on?

An 'Alarm' has not always been an inanimate device for informing people of a condition. 'Alarm' once was the condition in itself. People would be sent out to 'raise the alarm'. They wouldn't 'come on', but rather, 'go off' to do this.