Escapar is one of those verbs that can be used with or without the reflexive pronoun (let's call them "optionally pronominal verbs" for short), with only a subtle change in meaning.
Your first example sentence is... weird. If you mean that "his breath comes out of his mouth without him being able to prevent it" (as in, "he's so much in need of breath that he's panting hard and has to continue doing so for a while"), then the verb escaparse (pronominal escapar) is the right choice. But you would use it like this:
Se le escapa la respiración.
This construction has a sympathetic dative. By mentioning the third person patient (le = a él/ella) you make the possessive redundant (thus you don't say su respiración but la respiración). It's exactly parallel to more familiar phrases that associate a person with a possession or something closely related to him/her, especially body parts, such as
Se le enfrían los pies.
Se nos pierden las llaves.
Se me duermen las piernas.
This construction doesn't only exist for pronominal verbs but it's very common with some of them, because of reasons having to do with the fact that most are intransitive and have patientive subjects (subjects that experience a change or action passively).
You can place the subject in its more typical position at the beginning, too; it's a matter of subtly different emphases.