We present exact analytical solutions for topological edge states, including zero modes and in-gap states, in one-dimensional non-Hermitian quadripartite Su-Schrieffer-Heeger lattices under open boundary conditions. From these solutions, we obtain explicit asymptotic expressions for the edge-state energies and wave functions, and establish rigorous criteria governing their existence and spatial localization. Our results reveal how and under what conditions the non-Hermitian skin effect suppresses or delocalizes edge states through parameter-dependent decay lengths. Furthermore, we introduce for such systems a unified set of topological invariants that not only classify all edge states and restore the bulk-boundary correspondence, but also predict their spatial localization directions. This work thus paves the way for understanding and engineering topological phenomena in a broad class of non-Hermitian multipartite systems, even including those lacking chiral symmetry.

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• Edge states
• Topological effects in photonic systems
• Topological phase transition
• Topological phases of matter
• Topological insulators