A method is presented for automated photometric classification of supernovae (SNe) as Type Ia or non-Ia. A two-step approach is adopted in which (i) the SN light curve flux measurements in each observing filter are fitted separately to an analytical parametrized function that is sufficiently flexible to accommodate virtually all types of SNe and (ii) the fitted function parameters and their associated uncertainties, along with the number of flux measurements, the maximum-likelihood value of the fit and Bayesian evidence for the model, are used as the input feature vector to a classification neural network that outputs the probability that the SN under consideration is of Type Ia. The method is trained and tested using data released following the Supernova Photometric Classification Challenge (SNPCC), consisting of light curves for 20 895 SNe in total. We consider several random divisions of the data into training and testing sets: for instance, for our sample D-1 (D-4), a total of 10 (40) per cent of the data are involved in training the algorithm and the remainder used for blind testing of the resulting classifier; we make no selection cuts. Assigning a canonical threshold probability of p(th) = 0.5 on the network output to class an SN as Type Ia, for the sample D-1 (D-4) we obtain a completeness of 0.78 (0.82), purity of 0.77 (0.82) and SNPCC figure of merit of 0.41 (0.50). Including the SN host-galaxy redshift and its uncertainty as additional inputs to the classification network results in a modest 5-10 per cent increase in these values. We find that the quality of the classification does not vary significantly with SN redshift. Moreover, our probabilistic classification method allows one to calculate the expected completeness, purity and figure of merit (or other measures of classification quality) as a function of the threshold probability p(th), without knowing the true classes of the SNe in the testing sample, as is the case in the classification of real SNe data. The method may thus be improved further by optimizing p(th) and can easily be extended to divide non-Ia SNe into their different classes.