Single-photon avalanche diodes (SPADs) are novel image sensors that record photons at extremely high sensitivity. To reduce both the required sensor area for readout circuits and the data throughput for SPAD array, in this paper, we propose a snapshot compressive sensing single-photon avalanche diode (CS-SPAD) sensor which can realize on-chip snapshot-type spatial compressive imaging in a compact form. Taking advantage of the digital counting nature of SPAD sensing, we propose to design the circuit connection between the sensing unit and the readout electronics for compressive sensing. To process the compressively sensed data, we propose a convolution neural-network-based algorithm dubbed CSSPAD-Net which could realize both high-fidelity scene reconstruction and classification. To demonstrate our method, we design and fabricate a CS-SPAD sensor chip, build a prototype imaging system, and demonstrate the proposed on-chip snapshot compressive sensing method on the MINIST dataset and real handwritten digital images, with both qualitative and quantitative results.