Most homeowners liability insurance policies do in fact cover situations like you describe. Whether it was a dog bite or say you had a visitor to your home and they fell and injured themselves. This is called personal injury protection. However, it depends on the specific language of your homeowners policy. It may not mention specifically a "dog bite", but it probably mentions accidents suffered by visitors. I assume that the child that got bit by the dog was a visitor. Note that a possible area of contention may be wording in the policy that says something to the effect that you, as homeowner, agree to maintain a safe environment. It could be argued that the dog posed a unsafe environment.
Nevertheless, find your policy and read it, it should tell you what exactly is covered and what is not. Some of these policies are tricky and depending on how it is worded will determine if the accident will be covered or not.