Personally at 13 months if she doesn't have a paci, I wouldn't start that habit now - just my opinion of course.
Sounds like a habit. You can try wake to sleep: Set your alarm for one hour earlier then she usually wakes up - so in your case 12am. Go into her room quietly and gently stir her from a deep sleep. This should restart her sleep cycle and hopefully she won't wake at 1am. Be sure to not wake her completely. You just want to stir her. After you stir, personally I find it is better if you wake in te room silently for a few moments to give her a chance to go back into a good sleep. I find sometimes when I exit immediately, because he is in a light sleep, my leaving wakes him completely
So, just hang tight for a min or two after you stir her so she can settle, then go out and pray to the Sleep Gods.
The other thing you can try is the water down method. You can every 2-3 night add less milk and more water to the bottle in hope that when its all water she won't wake for it anymore. Here is a link about it, this mom used the method for a younger baby, however I can't see why it wouldn't work for an older one. You could also just cut back the amount of milk, replacing it with nothing and see if that works too. That's how I weaned the DF, every 2 night I put 2 ounces less in the bottle and that was that.
So, there are a few options for ya! Hope one of them is helpful. Be sure too she is adequately (as she can be) medicated for the teething.
Hugs, good luck,
Zoe