The answer is yes, but the landlord won't agree. That's the problem.
Legally, the landlord by not exterminating the rats has breached the implied warranty of habitability that is automatically part of your rental contract. Once one party to a contract breaks the contract, then the other party has no more obligation to adhere to the contract. Does that make sense? If your landlord breaks the contract, it's broken... you don't have to do anything more (like pay rent).
Probably what will happen though is that the landlord will withhold your security deposit and last month's rent. You will have to sue him in small claims court to get that money back. If you can't afford that, then keep calling the authorities.
Have you seen rats running about? Or just heard them? Have you seen their droppings or seen that they have been eating your food? No one should have to live with rats in the house, in any case, but there is "annoying" and there is "repulsive."
You should see an attorney who handled tenant cases on a contingency basis. You have a strong action for breach of the warranty of habitability because (1) you contacted management (2) you contacted authorities. I assume the authorities produced some sort of notice to management? Because they have not addressed the issue you are moving out - this is called "constructive eviction" and is also a cause of action for an affirmative suit. Your lawyer will then pile on terms like quiet enjoyment, emotional distress, pain and suffering, unlawful business practices, and the like.
If they don't comply with the security deposit law, you can add on penalties for failing to do so.