With the skin broken there is a possibility it will infect. Most simple cuts will heal fine on their own. If it forms a lump with redness around it then you'll need to get antibiotics as a life-saving measure.
The most common causes for this kind of thing that I can think of (remember I am not an expert or vet) in the order of occurrence.
1. Ringworm. This can be treated by treating the chinchilla every day with a medicated dust bath. Usually it starts on the nose and face but it can start at the base of the tail where the fur is not as dense. Here's how to treat that:
http://www.chinchillaclub.com/forum/index.php/topic,3994.0.htmlAlso I recommend cleaning the cage every day until it's cleared up. Ringworm is very contagious and can re-infect the chinchilla if it is on the cage or accessories. There are other types of parasitic fungi that have similar symptoms but don't always form the red pattern characteristic of ringworm.
2. Dry skin. This can be caused by dusting too frequently or coming into contact with urine in the cage. Usually the feet are effected as well. I've seen this happen occasionally with owners who use Carefresh bedding or have no bedding in the cage. If it's this you can expect the feet to look red and irritated and scaly. Having pine chips piled up where the chin can lay in it can help. Dust only once every other day (at most) and make sure the floor is scrubbed and free of urine residue.
4. Mechanical abrasion or trauma from something in the cage that it's rubbing against.
5. Neurotic behavior that some chinchillas seem to have causing fur chewing. Here's more on that:
http://www.chinchillaclub.com/forum/index.php/topic,3993.0.html6. Irritation from something the animal is allergic to coming into contact with the area. Usually the feet are affected as well. Sometimes this can be caused by food allergies but it's rare.
7. Fleas or mites which is very rare in chinchillas but can happen in individuals with low fur quality. You can look closely with a magnifying glass to see if you can see any.
Also now that the skin is reddened would you mind posting another picture? Perhaps someone who has experienced and seen this they might post. Also because you are seeing an escalation in these symptoms be very careful of food and water intake. If your chin goes off it's food you need to see a vet.
I hope this helps.