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Quantum transport between distant nodes that is robust to experimental imperfections is essential for quantum information processing. Here we experimentally demonstrate efficient and robust edge-to-edge transport of atomic momentum states in a synthetic lattice of Bose-Einstein condensate, simulating a dynamically modulated Su-Schrieffer-Heeger (SSH) model. This transport process relies on continuously controlling the effective nearest-neighbor couplings in the synthetic lattice, which constructs a unique chain between the left- and right-edge states. The robustness of such transport is protected by the chiral symmetry of the system, demonstrated by subjecting the lattices to coupling-strength disorders. Furthermore, we implement a splitter operation through an SSH model with a topological interface at its center. Our approach provides an efficient single operation to achieve robust momenta transport with potential applications in coherent quantum control in atom optics.