Mass Effect 3 on Xbox logs into Origin every time I start the game up, but I never actually gave it my Origin credentials, and (understandably) it does not seem to be associated with my Origin account. How do I make this association or access the new account it apparently created for me?
First of all, do you already have a Bioware Social Network account? If so, then this is also your EA/Origin account. You should go to this Account Help page and check to see which of these situations apply to you.
So in your case, if you've already connected to EA.com with another EA game on your 360, then they already know that that specific gamertag is associated with that specific EA account and you don't need to reassociate the identity.
However, you said that your console doesn't seem to be associated with your Origin account, but I would doublecheck this by logging into BSN and going to the User Entitlements page to see if your games+DLC are registered or not. If you end up having more than one BSN/EA account, then you have to figure out merging them (which is a whole other mess), which is also detailed on the Account Help page.
If you are sure your BSN/EA accounts are correctly set up, then the following is apparently the easiest way to do this from your console.
From the EA Support article here: