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The S = 1/2 honeycomb lattices have long been recognized as a potential host for a wide spectrum of appealing magnetic ground states. Of particular interest is the so-called valence bond solid state with singlet-triplet excitations (triplons), which is possibly realized in the delafossite-derived compound Cu₅SbO₆. Magnetic susceptibility measurements on single crystals indicate that Cu₅SbO₆ exhibits a spin gap with an average energy gap of 16 meV. To investigate the magnetic excitations in more details, we performed inelastic neutron scattering experiments on both powder and single-crystal samples. The results clearly show three magnetic excitation branches two of which are non-dispersive triplon bands. The existence of the two triplon bands with different energy gaps is ascribed to the presence of the Dzyaloshinski-Moriya (DM) interaction between the next-nearest-neighbor Cu²⁺ ions in the hexagon. The coupling between nearest-neighbor Cu²⁺ ions is likely ferromagnetic and a center of inversion between the nearest-neighbor spins prohibits the DM interaction. It is found that the z-component of the DM interaction splits the triplet states with m_s = 0 and m_s = ±1. At 2.8 K, the triplet states with m_s = ±1 is associated with the excitation at 15 meV and those with m_s = 0 with 18 meV. In addition to the non-dispersive bands, a relatively dispersive branch is also visible, which could be due to a non-vanishing dimer-dimer interaction.