This research focuses on the theory of harmonic univalent functions, a significant branch of complex analysis. The primary objective of this paper is to introduce a novel convolution operator incorporating the Rabotnov function and to utilize it to investigate the relationships between various subfamilies of harmonic functions defined in the open unit disk. The Rabotnov fractional exponential function, which serves as a generalization of the Mittag-Leffler function, is a crucial tool in mathematical modeling for material sciences, particularly in viscoelasticity. By applying a specific normalization to the Rabotnov function to obtain ℝ_K,β(z), we define a new linear operator F that acts on the analytic and co-analytic parts of harmonic mappings. The study establishes precise conditions under which the operator F maps functions from the classes of harmonic starlike functions S ^*,0_ℋ and harmonic convex functions K⁰_ℋ into the general class W_ℋ(Υ, α, Φ). These findings are derived through rigorous coefficient inequality analysis and the application of fundamental lemmas regarding harmonic mappings. Furthermore, the paper provides several illuminating corollaries and illustrative examples to demonstrate the geometric implications of the results.