I gargled with chlorhexidine gluconate for a bad sore throat. Fell asleep for about an hour and my throat feels a lot better. Is this normal?

Killed germs. Cholohexidine is an antimicrobial. This means that it will kill germs in your mouth when you gargle with it. That is why you feel better.
Yes. Yes because the chlorhexidene is a potent antibacterial agent and has helped reduce the surface populations of bacteria in your mouth and your throat. This reduction in bacteria results in less irritation to these tissues, so they feel better.