We report the first measurement of exclusive cross sections for $e^+e^-\to B\bar{B}$, $e^+e^-\to B\bar{B}^{*}$, and $e^+e^-\to B^{*} \bar{B}^{*}$ in the energy range from $10.63$~GeV to $11.02$~GeV. The $B$ mesons are fully reconstructed in a large number of hadronic final states, and the three channels are distinguished using the beam-energy-constrained mass variable. For each channel, the cross section shows an oscillatory behavior, with multiple maxima and minima. The final results on $\Upsilon(5S)\to \Upsilon(1S,2S)\eta$ and $\Upsilon(5S)\to \Upsilon(1S)\eta^{\prime}$ branching fractions are also presented. All these results are based on data collected by the Belle experiment at the KEKB asymmetric-energy $e^+e^-$ collider.