Lattice gauge theories are fundamental to various fields, including particle physics, condensed matter, and quantum information theory. Recent progress in the control of quantum systems allows for studying Abelian lattice gauge theories in table-top experiments. However, several challenges remain, such as implementing dynamical fermions in higher spatial dimensions and magnetic field terms. Here, we map D-dimensional U(1) Abelian lattice gauge theories onto qudit systems with local interactions for arbitrary D. We propose a variational quantum simulation scheme for the qudit system with a local Hamiltonian, that can be implemented on a universal qudit quantum device as the one developed in [Nat. Phys. 18, 1053-1057 (2022)]. We describe how to implement the variational imaginary-time evolution protocol for ground state preparation as well as the variational real-time evolution protocol to simulate non-equilibrium physics on universal qudit quantum computers, supplemented with numerical simulations. Our proposal can serve as a way of simulating lattice gauge theories, particularly in higher spatial dimensions, with minimal resources, regarding both system sizes and gate count.