Sparse Matrix-matrix Multiplication (SpMM) and Sampled Dense-dense Matrix Multiplication (SDDMM) are important sparse operators in scientific computing and deep learning. Tensor Core Units (TCUs) enhance modern accelerators with superior computing power, which is promising to boost the performance of matrix operators to a higher level. However, due to the irregularity of unstructured sparse data, it is difficult to deliver practical speedups on TCUs. To this end, we propose FlashSparse, a novel approach to bridge the gap between sparse workloads and the TCU architecture. Specifically, FlashSparse minimizes the sparse granularity for SpMM and SDDMM on TCUs through a novel swap-and-transpose matrix multiplication strategy. Benefiting from the minimum sparse granularity, the computation redundancy is remarkably reduced while the computing power of TCUs is fully utilized. Besides, FlashSparse is equipped with a memory-efficient thread mapping strategy for coalesced data access and a sparse matrix storage format to save memory footprint. Extensive experimental results on H100 and RTX 4090 GPUs show that FlashSparse sets a new state-of-the-art for sparse matrix multiplications (geometric mean 5.5x speedup over DTC-SpMM and 3.22x speedup over RoDe).