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I'm going to give you an intuitive answer. Keep in mind, this is not the "actual" answer, as the Hawking radiation is quite a bit more complex than the typical pop-sci explanation with virtual particles. But some intuitive justification is possible nevertheless.

I don't see how this event contributes to evaporation of the black
hole (, since the particles do not originate from the black hole).

You're missing a key point here.

When the pair was generated, those were virtual particles. After one side of the pair was absorbed by the black hole, and the other side was released, the released part is a real particle. Huge difference there - virtual vs real.

Virtual particles don't really exist the same way that you and me exist. They seem to exist for a very short time; the more energetic they are, the shorter the interval of their virtual "existence", per the Heisenberg equation. In many ways they are just a mathematical trick.

Think of the vacuum, where no real particles exist. Before, it's just vacuum. Right now, a virtual pair flickers briefly, then it's gone. In the future, it's vacuum again.

What was the energy before? Zero. What is the energy in the future? Zero. What's the energy during the flicker? Well, it basically zero, within the limits permitted by Heisenberg's equations. Bottom line is, virtual particles come and go, and they do not contribute to the energy balance of some empty chunk of space.

(I am ignoring here the concept of vacuum energy, for the sake of an intuitive explanation.)

But let's say one of the virtual particles gets trapped by the black hole, so it cannot annihilate with its counterpart. The other particle flies off in the opposite direction and escapes the black hole. What's worse, this is now a real particle - we've exceeded the duration permitted by the Heisenberg equations, so the one that escapes is not virtual anymore.

How did that particle become real?

This is a big issue, because virtual particles don't require an energy budget to briefly exist, while real particles do carry energy forever. Something prevented the virtual pair from annihilating itself, and boosted one of the components to the status of real particle. The virtual pair has zero energy. The real particle that gets away has non-zero energy. That energy has to come from somewhere.

It comes from the black hole. The black hole gives up some of its mass / energy (same thing) to boost one particle from virtual to real. The other particle is captured - but being virtual anyway, it doesn't really matter.

What this intuitive explanation doesn't say is how the boost actually happens. I dunno, magic. Somehow one of the virtual particles gets a chunk of energy from the black hole and becomes real.

Again, this is not the actual process. The actual process is more complex. This is just a pop-sci fairy tale.

EDIT: To hit closer to home, Hawking radiation is more like a close relative to the Unruh effect. Say an inertial observer sees empty space here in this chunk of volume. An accelerating observer would not see empty space in the same volume, but instead would see blackbody radiation. That's the Unruh effect.

Well, gravity and acceleration are the same thing, per general relativity. So the strong gravity near a black hole is equivalent to strong acceleration. Something similar to the Unruh effect must happen there. That's the Hawking radiation.

http://backreaction.blogspot.com/2015/12/hawking-radiation-is-not-produced-at.html

EDIT2: The other answers currently on this page provide useful alternative points, so check them out too.

These lecture notes address the issues to some degree, especially on slides 33-35.

Because in the strongly warped spacetime near the
horizon, virtual particles made from vacuum fluctuations
turn out to have negative energy density.

Energy density = energy per unit volume.

These particles indeed have positive mass -- look at the
one that escaped! -- but their mass is distributed very
strangely over spacetime. (Quantum-mechanically
speaking, particles have nonzero volume; this is an aspect
of the wave-particle duality.)

Matter with negative energy density is generally called
exotic matter

and, a bit later:

Quantum mechanical vacuum fluctuations in flat
spacetime - far from any strong gravitational field -
always have zero net energy density; they can never be
exotic.

However, in warped spacetime, vacuum fluctuations are
in general exotic: their net energy density is negative,
according to a distant observer measuring the energy
density by observation of the deflection of light by the
ensemble of fluctuations. The stronger the curvature, the
more negative the energy density looks.

This is the best explanation I have seen so far.

Heisenberg's principle allows you to temporarily violate energy conservation laws (e.g. create pairs of particles out of nothing) as long as you repay everything in time. The larger the particle-antiparticle pair, the quicker it has to be repaid. Converting a virtual pair to a real pair can be seen as generating a bit of negative-energy "exotic matter" (whatever that is) to represent the unpaid debt. Its energy is equal in size to the pair with the opposite sign. This then falls into the black hole along with one of the particles, decreasing the mass of the black hole overall.

The horizon of the black hole gets in the way of recombining some virtual pairs, so these conversions virtual->real will happen.

I found this lecture with the same idea (more detailed and less butchered):
http://teacher.pas.rochester.edu/Ast102/LectureNotes/Lecture19/Lecture19.pdf

I don't know if the experts will agree with this description, but here is how I understand it:

Both space and the event horizon are in constant quantum fluctuation. Essentially, the event horizon has tiny ripples. At points where the event horizon ripples up (above the average radius of the black hole), it has an above average amount of local energy. The intense gravity rapidly pulls that local bump back down, the falling bump sends that local energy concentration back across the rest of the event horizon.

Now let's consider possible virtual particle-pairs near the hole. If a stationary virtual particle-pair appears just above the event horizon, it will either recombine and vanish or the entire thing gets pulled into the hole and vanishes in zero. We need a virtual particle-pair that has an apparent motion away from the black hole, at nearly the speed of light. If that that virtual particle-pair is going fast enough to completely escape, they recombine and vanish. Zero net effect. We need a virtual particle-pair that is moving away from the black hole at nearly the speed of light, and we need a ripple in horizon which only catches one virtual particle. I believe the ripple must be under extreme downwards acceleration for it to pull away from the second virtual particle, to avoid catching both. And here's the key part: The energy-debt between the particle pair intensely pulls them towards each other. The trapped particle is being pulled upwards, effectively pulling upwards on the horizon that trapped it. This slows the fall of horizon-ripple, diminishing the energy that the falling ripple returns to the rest of the black hole.

The energy required to pull the two virtual particles apart equals the combined energy of the two non-virtual particles. So the falling ripple loses energy equal to two particles, and the hole eats one particle. Everything balances out with the one escaped particle.

I believe it works the same, regardless of whether the virtual particles are photons or a matter-antimatter pair.