Well... I was just about to type a message about how either myself or my phone must have been hallucinating earlier, because I haven't been able to get it to work again since, despite trying many times. Then, just before I started typing I said, "Okay Google" and sure enough my screen turned on and Google Now popped up. The next word out of my mouth was "Wow...", and Google was kind enough to pull up a link to World of Warcraft.
I've figured it out, though. Don't ask me why, but it works if the phone is plugged in. I'm sitting at my desk right now with the phone charging on a USB cable that's plugged into my desktop, and it works every time. I unplug it, turn the screen off and try again and it fails, every time.
I can't really think of much logic behind doing it this way. I wonder if it was intentional.