We present a lattice-QCD calculation of the B→πℓν semileptonic form factors and a new determination of the CKM matrix element textbarVubtextbar. We use the MILC asqtad (2+1)-flavor lattice configurations at four lattice spacings and light-quark masses down to 1/20 of the physical strange-quark mass. We extrapolate the lattice form factors to the continuum using staggered chiral perturbation theory in the hard-pion and SU(2) limits. We employ a model-independent z parametrization to extrapolate our lattice form factors from large-recoil momentum to the full kinematic range. We introduce a new functional method to propagate information from the chiral-continuum extrapolation to the z expansion. We present our results together with a complete systematic error budget, including a covariance matrix to enable the combination of our form factors with other lattice-QCD and experimental results. To obtain textbarVubtextbar, we simultaneously fit the experimental data for the B→πℓν differential decay rate obtained by the BABAR and Belle collaborations together with our lattice form-factor results. We find textbarVubtextbar=(3.72±0.16)×10-3, where the error is from the combined fit to lattice plus experiments and includes all sources of uncertainty. Our form-factor results bring the QCD error on textbarVubtextbar to the same level as the experimental error. We also provide results for the B→πℓν vector and scalar form factors obtained from the combined lattice and experiment fit, which are more precisely determined than from our lattice-QCD calculation alone. These results can be used in other phenomenological applications and to test other approaches to QCD.