How does reason justify the application, exactly?
Great question. Here I would say that this is where the study of nature comes in. You look at the universe and derive from what you see that it is natural and has no gods over it nor needs any outside justification. You realize that your consciousness is available to you but for a short time. You realize that being alive without pain is "good" and a privilege and a pleasure that is available only for a while. You decide based on all the facts that being alive is something that feels right AND you consciously desire to continue it and to make the best of it.
All sorts of reasoning can be used from all sorts of direction to reinforce this, but in the end the assessment you make of the universe and your place is that it is desirable - maybe as Nietzsche might say you say "Yes" to existence and the world, and you choose to value every moment of it that is available to you. Lots of Epicurean sayings reinforce that positive outlook on life. That's the direction I would take that answer.
You just have to tune your mind in to the small pleasures that your body is experiencing when it is just operating normally.
And as Patrikious says in using the word "tune" several times, the decision to embrace this worldview, rather than fear and otherworldliness and escape, is a conscious decision that requires you to "tune in" to this perspective. It doesn't happen just by falling off the turnip truck or by simply "observing" the light of day -- you have to put it together in your mind in a way that makes sense.