Triangle counting is a fundamental graph algorithm used to identify the number of triangles within a graph. This algorithm can be reformulated into linear algebraic operations, including sparse matrix multiplication, intersection and reduction. Modern GPUs, equipped with Tensor Cores, offer massive parallelism that can significantly accelerate graph algorithms. However, leveraging Tensor Cores, originally designed for dense matrix multiplication, to handle sparse workloads for triangle counting presents non-trivial challenges. In this paper, we conduct an in-depth analysis of the state-of-the-art techniques that utilizes Tensor Cores for matrix operations, identifying critical performance shortfalls. Based on these insights, we introduce ToT, which enhances the utilization of Tensor Cores and expands their functionalities for diverse sparse matrix operations. In experiments, ToT is evaluated against state-of-the-art methods. ToT outperform the second-fastest method with an 11.56X speedup in end-to-end execution. Also, it achieves up to 32.31X memory savings. This work represents a pioneering exploration into utilizing Tensor Cores for accelerating graph algorithms.