CNN on python with Keras
You can train that almost with the same architecture (of course it depends on this architecture, if it is already bad then it will not be useful on more classes too. I would suggest to use the state of the art model architecture for dogs and cats classification) but you will also need the dogs and cats dataset in addition to this third class dataset. Unfortunately, it is not possible to use pre-trained for making predictions between all 3 classes by only training on the third class later. 
So, cut to short, you will need to have all three datasets and train the model from scratch if you want to make predictions between these three classes otherwise use the pre-trained and after training it on third class it can predict if some image belongs to this third class of not.
I made a simple CNN that classifies dogs and cats and I want this CNN to detect images that aren't cats or dogs and this must be a third different class. How to implement this? should I use R-CNN or something else?
P.S I use Keras for CNN