The rapid growth in the size of large language models has necessitated the partitioning of computational workloads across accelerators such as GPUs, TPUs, and NPUs. However, these parallelization strategies incur substantial data communication overhead significantly hindering computational efficiency. While communication-computation overlap presents a promising direction, existing data slicing based solutions suffer from tail latency. To overcome this limitation, this research introduces a novel communication-computation overlap technique to eliminate this tail latency in state of the art overlap methods for distributed LLM training. The aim of this technique is to effectively mitigate communication bottleneck of tensor parallelism and data parallelism for distributed training and inference. In particular, we propose a novel method termed CommFuse that replaces conventional collective operations of reduce-scatter and all-gather with decomposed peer-to-peer (P2P) communication and schedules partitioned computations to enable fine-grained overlap. Our method provides an exact algorithm for reducing communication overhead that eliminates tail latency. Moreover, it presents a versatile solution compatible with data-parallel training and various tensor-level parallelism strategies, including TPSP and UP. Experimental evaluations demonstrate that our technique consistently achieves lower latency, superior Model FLOPS Utilization (MFU), and high throughput.