What do you call someone who always talks a situation in their favor?

The phenomenon you speak of has a distinct studied name which I remember from a psych class from long ago - ad hoc hypothesizing. If I remember correctly it is when you compensate for abnormalities or issues by creating false rationalization(s) about a topic - in basic terms, changing your reaction based on outcome. I would just call them a liar. These people are certainly pathological.

The thing about ad hoc hypothesis is that they are stated in a way that is almost impossible to prove wrong. In your example Mary states she purposely left the food in her teeth. It is almost impossible to discredit this.

Now if the person is just doing it on purpose to be funny they would be a smart-ass.

Also for a person who is kind of in-between the liar and smart-ass, the person who change an argument based on the end result would have circular logic. In your specific example this could work, but we know that 99% of people don't leave meat in their teeth on purpose.

Tendentious means, in the definition given by Oxford Dictionaries Online, ‘expressing or intending to promote a particular cause or point of view, especially a controversial one’. It is, however, normally reserved for weightier matters. You may therefore have to rely on something more basic, such as lying, in cases like this.

It depends of course on the register of language. Informally, it could be called a smooth talker.

In principle, that should apply specifically to cases when the person wants another to do something; but Urban Dictionary seems (as I would) to use it in a broader sense.