Multiclass classification with Neural Networks
Indeed, this is the standard interpretation of continuous classifier outputs, not only for neural networks, but for the more general case called Softmax Regression.

Thus, provided that you have used softmax activation on the final layer (in order, among other things, to ensure that your outputs indeed sum up to 1), you can interpret the continuous outputs as the respective probabilities of a particular data sample belonging to each one of your classes.

See also the discussion in this (rather unfortunately titled) discussion at SO: 

How to convert the output of an artificial neural network into probabilities?