A star coloring of a graph $G$ is a proper vertex coloring such that no path on four vertices is bicolored. The smallest integer $k$ for which $G$ admits a star coloring with $k$ colors is called the star chromatic number of $G$, denoted as $χ_s(G)$. In this paper, we study the star coloring of tensor product of two graphs and obtain the following results. 1. We give an upper bound on the star chromatic number of the tensor product of two arbitrary graphs. 2. We determine the exact value of the star chromatic number of tensor product two paths. 3. We show that the star chromatic number of tensor product of two cycles is five, except for $C_3 \times C_3$ and $C_3 \times C_5$. 4. We give tight bounds for the star chromatic number of tensor product of a cycle and a path.