Theory 1: moisture
Were the leaves wet at all? I worked in a restaurant before, tending to the fridge. I had to make sure the salads were washed and thoroughly spun. Wet greens wilted and rotted faster and we had to toss them. The roots can be wet and exposed to water, but make sure the leaves stay dry.
Theory 2: freezer burn
This link mentions that cilantro can be damaged by freezing temperatures, which could be an explanation for what you're seeing? Is it so cold in your area that your refrigerator actually dips into freezing temps because the air around it does?
Meats can get freezer burn or taste off after freezing. One explanation I've read is that water freezes as sharp crystals in and around the cells. Upon defrosting, these sharp crystals can cut through cell membranes, causing cell contents to leak out and alter the meat structure. In theory the same could apply to anything organic.
Theory 3: dehydration
The link above and others I've read suggest storing cilantro as you do in the fridge, but covered with a plastic bag. Some wrap their cilantro in a paper towel. Either way, this creates an enclosed or somewhat-enclosed space that slows down moisture escaping from the leaves. The air in the fridge is generally dry and can suck moisture out of the leaves. I know this sounds like the opposite of too much moisture, but there's a window for how much moisture plants want--neither too little nor too much is desired.
I'd try to resolve this in the order above. If you left the leaves wet, I'd say that's a very likely culprit. I'd check your fridge temperature next: you could put an ice tray near where you put the cilantro to see if it freezes or starts freezing overnight. If it's still wilting fast, try doing what you did with a plastic or paper bag over the leaves--making sure the bag isn't crushing them.