The Gamma function Γ(z) is fundamental in analysis and fractional calculus; however, its poles at the negative integers prevent its direct use as an integral transform kernel. In this paper, we introduce the Gamma transform, a novel analytic operator derived from a distributional decomposition of Γ(z). By representing the Gamma function as a convergent series of complex delta functionals, we obtain a regularized formulation that is free of singularities and well defined for real, fractional, and complex orders. The proposed transform exhibits structural properties analogous to classical integral transforms, including the Laplace and Mellin transforms, while incorporating discrete factorial-type weighting inherent to the Gamma function. Fundamental properties of the Gamma transform are established, including linearity, boundedness, inversion, and convolution theorems, ensuring consistency with standard operational calculus. This framework naturally extends classical transform theory to fractional settings without loss of analytic rigor. Applications to ordinary and fractional differential equations are presented to demonstrate the effectiveness of the Gamma transform in solving initial- and boundary-value problems, where it provides compact representations and inherent regularization. The results illustrate how the transform unifies classical analytic methods with modern fractional analysis. The Gamma transform offers a new, singularity-free tool for fractional calculus and related fields, contributing to the development of analytic techniques within the context of fractal and fractional systems.