In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.