We report the first measurement of the neutrino-oxygen neutral-current quasielastic (NCQE) cross section. It is obtained by observing nuclear deexcitation $\gamma$-rays which follow neutrino-oxygen interactions at the Super-Kamiokande water Cherenkov detector. We use T2K data corresponding to $3.01 \times 10^{20}$ protons on target. By selecting only events during the T2K beam window and with well-reconstructed vertices in the fiducial volume, the large background rate from natural radioactivity is dramatically reduced. We observe 43 events in the $4-30$ MeV reconstructed energy window, compared with an expectation of 51.0, which includes an estimated 16.2 background events. The background is primarily nonquasielastic neutral-current interactions and has only 1.2 events from natural radioactivity. The flux-averaged NCQE cross section we measure is $1.55 \times 10^{-38}$ cm$^2$ with a 68\% confidence interval of $(1.22, 2.20) \times 10^{-38}$ cm$^2$ at a median neutrino energy of 630 MeV, compared with the theoretical prediction of $2.01 \times 10^{-38}$ cm$^2$.