Something that is not obvious based on the description is that you need to land damage on the Owl for it to register. You can do this in a number of ways, like waiting until you get enough speed so that you hit first or building up enough armor that you can resist a hit, but these all take some patience, and you won't always get the Reprieve door in the later depths (nor would you want to potentially waste an near entire run for that).
The way I did this is using the Chaotic Underwear, which is a grey helmet that reverses the speed order, so if you're faster, you attack second, and if you're slower, you attack first. This will let you get the first hit in very early on. You'll definitely die, but you'll get the achievement too!