A new measurement of the $^{65}$Cu($\textit{γ}$, $\textit{n}$)$^{64}$Cu photoneutron cross section is performed using quasi-monoenergetic, tunable $\textit{γ}$-ray beams produced at the Shanghai Laser Electron Gamma Source (SLEGS). The energy spectrum of the SLEGS $\textit{γ}$-ray beams incident on the isotopically enriched $^{65}$Cu target is monitored using a BGO detector, while the photoneutron yields are determined with a moderated $^{3}$He detection array with high and flat efficiency. Within the energy range of $10.1 \le E_\gamma \le 17.6$ MeV, the measured $\sigma(E_\gamma)$ data have an uncertainty of $\lesssim 4$ %, and a pronounced giant-dipole peak is observed at $E_\gamma \simeq 16.65$ MeV with a maximal cross section of $\sigma_{\text{max}} \simeq 137$ mb. These photoneutron data are compared with previous experimental results and are employed to extract the $\textit{γ}$-ray strength function of $^{65}$Cu above the neutron threshold. Furthermore, we calculate the radiative neutron capture cross sections and astrophysical reaction rates for $^{64}$Cu, which is a short-lived intermediate nucleus whose reaction rate controls the local abundance distribution in the weak $\textit{s}$-process. It is found that the calculated $^{64}$Cu($\textit{n,}$ $\textit{γ}$)$^{65}$Cu data have an overall agreement with ENDF/B-VIII.0, JEFF-3.3, and TENDL-2023 evaluations, and the corresponding astrophysical reaction rates are consistent with those reported in the JINA REACLIB database.