Sampling from a categorical distribution is mathematically simple, but in large-vocabulary decoding, it often triggers extra memory traffic and extra kernels after the LM head. We present FlashSampling, an exact sampling primitive that fuses sampling into the LM-head matmul and never materializes the logits tensor in HBM. The method is simple: compute logits tile-by-tile on chip, add Gumbel noise, keep only one maximizer per row and per vocabulary tile, and finish with a small reduction over tiles. In tensor-parallel decoding, FlashSampling replaces the all-gather of logits with streaming peer-to-peer writes: This overlaps GPU-to-GPU communication with computation and HBM loads across up to 8 GPUs, with near-ideal scaling at large batch sizes. Our kernel is exact because argmax decomposes over partitions; grouped variants for online and tensor-parallel settings are exact by hierarchical factorization of the categorical distribution. FlashSampling demonstrates kernel-level speedups on decode workloads across 4 different datacenter GPUs (H100, H200, B200, B300), and in end-to-end vLLM experiments, it reduces time per output token by up to $10\%$ on the models we test. These results show that exact sampling, with no approximation, can be integrated into the matmul itself, consolidating the bandwidth-bound sampling step in an efficient epilogue.