On the one hand, I get irritated when someone takes 15 minutes of YouTube time to share 30 seconds worth of information; on the other hand, I'm grateful they took the time to share the information in the first place. It's not like they charged me for it.
In the video at the bottom of this entry, home improvement pro Paul Ricalde demonstrates a method of fixing holes in drywall that I've never seen before, which he calls the "butterfly patch" method. By looking at the photos below, you can probably figure it out without needing to sit through the vid.
Basically you take a hole saw larger than the hole in the drywall, and drill just partially through a scrap piece of drywall from the non-show side.
Then you cut a larger square around that circle. This part might require watching the video to grasp, but by cutting radiating lines in the paper from the circle's edges to the square's edges, you create trapezoidal chunks of drywall that you can break out, while leaving the paper front intact.
What's left over is the so-called butterfly.
You can make these whatever size you need. You'll then take the hole saw and cut a clean hole through a piece of wood you'll use as a guide. Here Ricalde's got two different sizes.
You place this template over the hole in the drywall, and use it to guide your hole saw as you cut out the damaged part.
Then you slather joint compound around the edges of the hole…
…take your butterfly…
..and smush it into the hole.
By using a taping knife to squeeze the joint compound out from beneath the paper, and mudding it all over…
…Ricalde quickly gets down to a flawless surface, with no visible trace of the repair.
Here's the video, if the photos didn't do it for you: