Background: The recent observation of a neutron-star merger finally confirmed one astrophysical location of the rapid neutron-capture process (r-process). Evidence of the production of A<140 nuclei was seen, but there is still little detailed information about how those lighter elements are produced in such an environment. Many of the questions surrounding the A≈80 nuclei are likely to be answered only when the nuclear physics involved in the production of r-process nuclei is well understood. Neutron-capture reactions are an important component of the r-process, and neutron-capture cross sections of r-process nuclei, which are very neutron rich, have large uncertainties. Purpose: Indirectly determine the neutron-capture cross section and reaction rate of Zn73(n,γ)Zn74. Methods: The nuclear level density (NLD) and γ-ray strength function (γSF) of Zn74 were determined following a total absorption spectroscopy (TAS) experiment focused on the β decay of Cu74 into Zn74 performed at the National Superconducting Cyclotron Laboratory. The NLD and γSF were used as inputs in a Hauser-Feshbach statistical model to calculate the neutron-capture cross section and reaction rate. Results: The NLD and γSF of Zn74 were experimentally constrained for the first time using β-delayed γ rays measured with TAS and the β-Oslo method. The NLD and γSF were then used to constrain the neutron-capture cross section and reaction rate for the Zn73(n,γ)Zn74 reaction. Conclusions: The uncertainty in the neutron-capture cross section and reaction rate of Zn73(n,γ)Zn74 calculated in TALYS was reduced to under a factor of 2 from a factor of 5 in the cross section and a factor of 11 in the reaction rate using the experimentally obtained NLD and γSF.