In this paper, a moment-based modification of the classical Bernstein operators is introduced. The proposed operators incorporate both a domain transformation and an adjustment by the second central moment to improve the approximation properties. We investigate their convergence behavior using tools such as the Lipschitz class and Peetre's $\kappa -$functional. Quantitative estimates are established based on the classical and second-order modulus of continuity. Furthermore, a Voronovskaja-type theorem is provided to analyze the asymptotic behavior. Theoretical results are supported by numerical examples and graphical illustrations that demonstrate the effectiveness of the operators.