We present the first unquenched lattice-QCD calculation of the hadronic form factors for the exclusive decay B¯→Dℓν¯ at nonzero recoil. We carry out numerical simulations on 14 ensembles of gauge-field configurations generated with 2+1 flavors of asqtad-improved staggered sea quarks. The ensembles encompass a wide range of lattice spacings (approximately 0.045 to 0.12 fm) and ratios of light (up and down) to strange sea-quark masses ranging from 0.05 to 0.4. For the b and c valence quarks we use improved Wilson fermions with the Fermilab interpretation, while for the light valence quarks we use asqtad-improved staggered fermions. We extrapolate our results to the physical point using rooted staggered heavy-light meson chiral perturbation theory. We then parametrize the form factors and extend them to the full kinematic range using model-independent functions based on analyticity and unitarity. We present our final results for f+(q2) and f0(q2), including statistical and systematic errors, as coefficients of a series in the variable z and the covariance matrix between these coefficients. We then fit the lattice form-factor data jointly with the experimentally measured differential decay rate from BABAR to determine the CKM matrix element, textbarVcbtextbar=(39.6±1.7QCD+exp±0.2QED)×10−3. As a byproduct of the joint fit we obtain the form factors with improved precision at large recoil. Finally, we use them to update our calculation of the ratio R(D) in the Standard Model, which yields R(D)=0.299(11).