We describe a search and classification procedure for gravitational waves emitted by core-collapse supernova (CCSN) explosions, using a convolutional neural network (CNN) combined with an event trigger generator known as Wavelet Detection Filter (WDF). We employ both a 1-D CNN search using time series gravitational-wave data as input, and a 2-D CNN search with time-frequency representation of the data as input. To test the accuracies of our 1-D and 2-D CNN classification, we add CCSN waveforms from the most recent hydrodynamical simulations of neutrino-driven core-collapse to simulated Gaussian colored noise with the Virgo interferometer and the planned Einstein Telescope sensitivity curve. We find classification accuracies, for a single detector, of over 95% for both 1-D and 2-D CNN pipelines. For the first time in machine learning CCSN studies, we add short duration detector noise transients to our data to test the robustness of our method against false alarms created by detector noise artifacts. Further to this, we show that the CNN can distinguish between different types of CCSN waveform models.