We report the analysis of the three-body e^{+}e^{-}→BB[over ¯]π^{±}, BB[over ¯]^{*}π^{±}, and B^{*}B[over ¯]^{*}π^{±} processes, including the first observations of the Z_{b}^{±}(10610)→[BB[over ¯]^{*}+c.c.]^{±} and Z_{b}^{±}(10650)→[B^{*}B[over ¯]^{*}]^{±} transitions that are found to dominate the corresponding final states. We measure Born cross sections for the three-body production of σ(e^{+}e^{-}→[BB[over ¯]^{*}+c.c.]^{±}π^{∓})=[17.4±1.6(stat)±1.9(syst)] pb and σ(e^{+}e^{-}→[B^{*}B[over ¯]^{*}]^{±}π^{∓})=[8.75±1.15(stat)±1.04(syst)] pb and set a 90% C.L. upper limit of σ(e^{+}e^{-}→[BB[over ¯]]^{±}π^{∓})<2.9 pb. The results are based on a 121.4 fb^{-1} data sample collected with the Belle detector at a center-of-mass energy near the ϒ(10860) peak.