Effective-source formulations provide a practical route to gravitational self-force (GSF) calculations when singular retarded fields cannot be handled directly mode by mode. Before this work, frequency-domain effective-source calculations of the first-order GSF had been performed only for circular Schwarzschild orbits. We construct and implement a frequency-domain extended effective-source (EES) formulation for first-order Lorenz-gauge GSF on eccentric orbits in Schwarzschild spacetime. The central obstacle is that radial libration forces the physical puncture and effective source to switch between interior and exterior branches, limiting their differentiability and producing slow spectral convergence. We overcome this obstruction by analytically extending both branches across the libration region and solving the coupled Lorenz-gauge perturbation equations for the resulting smooth extended sources. This provides an end-to-end gravitational EES implementation for eccentric motion and validates a computational ingredient needed for extending frequency-domain effective-source calculations toward second order.