Right. 私は犬に噛み殺された is more natural, though.
FYI, 私は犬で殺された can mean "I was killed by a dog", but it highly suggests the existance of doer(= killer) who ordered the dog to kill the subject in this sentence. Thus, the nuance is more close to "I was killed by a person who set a dog on me". 犬 is interpreted as the means to kill in this case. (Or, "I was killed by a person due to a dog problem" when 犬 is the cause/reason to be killed. 犬のことで殺された is more common in this case, though.)