We report the analysis of the three-body e+e-→BBπ±, BB∗π±, and B∗B∗π± processes, including the first observations of the Zb±(10610)→[BB∗+c.c.]± and Zb±(10650)→[B∗B∗]± 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∗+c.c.]±π)=[17.4±1.6(stat)±1.9(syst)] pb and σ(e+e-→[B∗B∗]±π)=[8.75±1.15(stat)±1.04(syst)] pb and set a 90% C.L. upper limit of σ(e+e-→[BB]±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 (10860) peak.