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Controlled breathing seems to be key to a lot of things, from dealing with adrenaline dumps and panic to acute and chronic pain. It (controlled breathing) is a key element of the Russian martial art of Systema, and altho I am not a Systema practitioner, at least formally, I have found its focus on breathing and relaxation to be very helpful. I think a key thing here is, like anything else, practice practice practice controlling your breathing in all kinds of situations so that it is a trained response when you hit the "Oh CRAP" moment.

The adrenaline dump, with increased heart rate and all that, is going to happen, and to some degree that is good, because it gets you prepped for battle (or fire suppression, or trauma first aid, or whatever). I think maybe more of what the OP was really getting into was the panic/freeze mode. At least one of the things that induces panic and freezing is not knowing what to do, which is why training as realistically as possible, with some stress involved, is a big help. If you've "been there" before, or at least "close to there," you are more likely to know what to do. Hence all the advice on training.

Rory Miller discusses panic and particularly the freeze response, with respect to being attacked, in his book Facing Violence. It is relatively inexpensive on Amazon, especially in Kindle form, highly recommend it. He goes into some detail as to why it happens, and some things you can do to try to break out of it. He notes that even very experienced fighters get caught by it, but tend to have figured out how to get out of it quickly. He is more focused on the instantaneous event, like the first clue you have that something is wrong is when someone smacks you on the head from behind and (providing you not knocked unconscious) you are instantly confused and frozen. However the extended instance where you know someone is in the house and likely coming for you can induce it as well, obviously, but having mentally and physically trained yourself can help you tamp the panic/freeze monster down.