They're taking some real ass liberties from the comics. Then again, they have to since they already killed Dale in season 2.
Everyone is already infected with the zombie disease, but it's only triggered when you die. Aside from that, you're still a walking talking human being. Now if you get bit by a zombie, the infection kills you - hence you turn into a zombie. People don't just walk around after they get bit and turn into a zombie. Usually there's a strong fever that kills you, etc.
So, cutting the infected area before it can travel through your body should work, like it does for many other diseases. It all depends how fast it goes through the blood stream.