Harry Potter help, please?

As I was rereading the Deathly Hallows, yet again for the 10th time, I tried desperately to ignore this nagging question forming at the back of my mind. After the conversation with Harry and Dumbledore, do you guys remember when there was a little creature kind of like "a baby," as Harry described it, huddling in a corner? Well, I figured later that it was Voldemort's horcux. What I don't get is how Voldemort killed the Horcrux inside Harry with merely a killing curse. I thought to destroy a horcrux, you need a weapon so powerful, like the sword of Gryffindor that was stained with basilisk blood, or basilisk fangs. I though a killing curse isn't enough. Please explain thoroughly in a clear way. Thanks.