The origin of GeV emission from the early epoch of gamma-ray bursts (GRBs) is a widely discussed issue. The long gamma-ray burst GRB 170405A, observed by the Fermi Gamma-ray Space Telescope, showed high-energy emission delayed by similar to 20 s with respect to the X-ray emission, followed by temporally fading gamma-ray emission lasting for similar to 1000 s, as commonly observed in high-energy GRBs. In addition, a high-energy spectral cutoff at similar to 50 MeV was detected in the prompt-emission phase. If this feature is caused by pair-production opacity, the bulk Lorentz factor of the GRB ejecta can be estimated to be Gamma(bulk) = 170-420. Simultaneously with Fermi, GRB 170405A was observed by the Swift/Burst Alert Telescope and X-ray telescope, and a clear optical onset was detected similar to 200 s after the burst by the Swift/Ultraviolet Optical Telescope. By coupling the deceleration time to the derived bulk Lorentz factor, the deceleration time was found to correspond to the delayed onset in the optical band. While the delayed onset in the optical band is evidence that this emission had an external shock origin, the temporally extended emission in the GeV band before the optical onset is hard to reconcile with the standard synchrotron emission from the same external shock. This may imply that the common feature of GeV emission with a power-law decay does not necessarily have the same origin as the optical afterglow in all GRBs detected by the Fermi/Large Area Telescope, particularly in their early epoch. Another mechanism to explain the GeV emission in GRB 170405A may be required, such as an internal shock or inverse Compton emission.