Hawking radiation has nothing to do with matter-antimatter annihilation. Mass and anti-mass are both forms of positive energy, and therefore matter and antimatter particles emit energy when they annihilate in accordance with conservation of energy. The particle-model of the quantum foam involved in Hawking-radiation consists of "virtual particles" with positive and negative energy (which is a bit odd, because permanent negative energy particles are not known to exist, but virtual particles in well accepted quantum field theories already break all sorts of rules that normal particles follow). These negative particles could conceivably be negative matter, or negative antimatter, or negative bosons (if you don't think bosons should be called matter). When equivalent positive and negative energy particles annihilate each other, it releases no energy, which is in accordance with conservation of energy because the net energy is zero. The way these particles can (at least theoretically) exist in vacuum is that their net energy always adds up to 0 in such situations, (or, if it turns out that vacuum actually has some minimum energy, it adds up to that).
I will tell you now that I don't actually understand the physics, but the basic idea behind Hawking radiation, as well as Unruh radiation, is that something about gravity/acceleration can change the behavior of what would otherwise be vacuum (from a different observer's perspective) so that it emits black-body radiation. The theory behind Hawking radiation then says that something about the nature of black holes creates a net effect where positive energy moves up (away from the black hole, i.e. the Hawking radiation) and (in accordance with conservation of energy) negative energy moves down, reducing the mass of the the black hole. (Note that the energy density of space is believed to be what curves spacetime, creating gravity. I suspect this somehow ends up always being "mass" in some sense because of relativity or virtual particles or something, but I really don't know.)