The answer to your question is the gravitational field of the black hole, which already exists before the (for example neutron star) collapses into a black hole and, moreover, this gravitational field extends to infinity (so outside the horizon).
Since the gravitational field already exists for the collapsing star, when the collapse "happens", the event horizon (which is not a physical thing, rather, a boundary) happens to be so that the gravitational field extends outside of it too.
A black hole, however, can have an electric charge, which means there is an electric field around it. This is not a paradox because a static electric field is different from electromagnetic radiation. Similarly, a black hole has a mass, so it has a gravitational field around it. This is not a paradox either because a gravitational field is different from gravitational radiation.
How does gravity escape a black hole?
The fact that the gravitational field extends outside the horizon, is no contradiction. Nothing, no information or particles need to travel from inside, because the gravitational field of the collapsing star (for example a neutron star) already existed outside the boundary (that we later call event horizon), and so the energy of this gravitational field can (and in your example does) influence the quantum fields outside the horizon.
This is what is used in the Hawking radiation arguments, where the virtual loop at the event horizon has one particle interacting with the gravitational field and energy is supplied by it so that the other becomes real and exits as real.
Can virtual particles be 'boosted' into becoming real particles by fields other than gravity?
Now there are many interpretations of Hawking radiation, in this one, the extreme energy of the gravitational field gives this "boost" to the quantum fields, helping fluctuations to manifest as (seen from a far away observer) as photons coming from the (outside region of the) black hole. This radiation is interpreted as Hawking radiation, and the reason it decreases the mass of the black hole is because it takes energy from the gravitational field of the black hole. No need for particles to go in or out the horizon or anything to be influenced from inside the horizon (of course there are many equivalently good interpretations, including particles going in and out but these all originate from outside the horizon), but just energy taken from the gravitational field of the black hole, decreasing its mass, which corresponds to the fact that in mainstream physics, the mass of the black hole is derived from its gravitational field's energy (ADM, Komar, Bondi).
So again, the answer is the gravitational field (and its energy) of the black hole which already exists outside the horizon. Please note that there is a way to view this intuitively (if I understand correctly that you are assuming an external view of the black hole) where all the energy of the black hole is outside the horizon (because it never crosses in a fininte amount of time), and it all evaporates before the black hole could fully form for a far away observer.