The rare decay B→πℓ+ℓ- arises from b→d flavor-changing neutral currents and could be sensitive to physics beyond the standard model. Here, we present the first ab initio QCD calculation of the B→π tensor form factor fT. Together with the vector and scalar form factors f+ and f0 from our companion work [J. A. Bailey et al., Phys. Rev. D 92, 014024 (2015)], these parametrize the hadronic contribution to B→π semileptonic decays in any extension of the standard model. We obtain the total branching ratio BR(B+→π+μ+μ-)=20.4(2.1)×10-9 in the standard model, which is the most precise theoretical determination to date, and agrees with the recent measurement from the LHCb experiment [R. Aaij et al., J. High Energy Phys. 12 (2012) 125].