To save a matplotlib plot to a HDF5 file, you can use the h5py library in Python. First, create an HDF5 file using h5py and then store the data from the matplotlib plot into the file. This can be achieved by converting the matplotlib plot into an image format, such as PNG, using the savefig() method, and then saving the image data to the HDF5 file. Make sure to import the necessary libraries and handle any exceptions that may arise during the process.