A search for the flavor-changing neutral-current decay B^{+}→K^{+}νν[over ¯] is performed at the Belle II experiment at the SuperKEKB asymmetric energy electron-positron collider. The data sample corresponds to an integrated luminosity of 63  fb^{-1} collected at the ϒ(4S) resonance and a sample of 9  fb^{-1} collected at an energy 60 MeV below the resonance. Because the measurable decay signature involves only a single charged kaon, a novel measurement approach is used that exploits not only the properties of the B^{+}→K^{+}νν[over ¯] decay, but also the inclusive properties of the other B meson in the ϒ(4S)→BB[over ¯] event, to suppress the background from other B meson decays and light-quark pair production. This inclusive tagging approach offers a higher signal efficiency compared to previous searches. No significant signal is observed. An upper limit on the branching fraction of B^{+}→K^{+}νν[over ¯] of 4.1×10^{-5} is set at the 90% confidence level.