This paper integrates a classic mel-cepstral synthesis filter into a modern neural speech synthesis system towards end-to-end controllable speech synthesis. Since the mel-cepstral synthesis filter is explicitly embedded in neural waveform models in the proposed system, both voice characteristics and the pitch of synthesized speech are highly controlled via a frequency warping parameter and fundamental frequency, respectively. We implement the mel-cepstral synthesis filter as a differentiable and GPU-friendly module to enable the acoustic and waveform models in the proposed system to be simultaneously optimized in an end-to-end manner. Experiments show that the proposed system improves speech quality from a baseline system maintaining controllability. The core PyTorch modules used in the experiments will be publicly available on GitHub.