We introduced a Szász–Durrmeyer-type family of positive linear operators generated by Euler polynomials of negative order $-k$, where $k\in \mathbb{N}$. The corresponding discrete kernel admits a finite binomial mixture of shifted Poisson probabilities, while the Durrmeyer integral part is described by a Gamma-type kernel. This Poisson–Gamma structure makes positivity and normalization transparent and provides a convenient basis for the computation of algebraic and central moments. Using these moments, we established quantitative estimates, weighted Korovkin-type convergence, and a compact-uniform Voronovskaya-type theorem. The asymptotic formula shows that the classical approximation order is preserved, whereas the negative-order Euler parameter affects only the drift term in the first-order asymptotic profile.