We present an analysis of exclusive charmless semileptonic B-meson decays based on 377 × 10^6 BB̅ pairs recorded with the BABAR detector at the Υ(4S) resonance. We select four event samples corresponding to the decay modes B^0 → π^-ℓ^+ν, B^+ → π^0ℓ^+ν, B^0 → ρ^-ℓ^+ν, and B^+ → ρ^0ℓ^+ν and find the measured branching fractions to be consistent with isospin symmetry. Assuming isospin symmetry, we combine the two B → πℓν samples, and similarly the two B → ρℓν samples, and measure the branching fractions B(B^0→π^-ℓ^+ν)=(1.41 ± 0.05 ± 0.07) × 10^(-4) and B(B^0 → ρ^-ℓ^+ν)=(1.75 ± 0.15 ± 0.27) × 10^(-4), where the errors are statistical and systematic. We compare the measured distribution in q^2, the momentum transfer squared, with predictions for the form factors from QCD calculations and determine the Cabibbo-Kobayashi-Maskawa matrix element |V_(ub)|. Based on the measured partial branching fraction for B → πℓν in the range q^2 < 12 GeV^2 and the most recent QCD light-cone sum-rule calculations, we obtain |V_(ub)|=(3.78 ± 0.13^(+0.55)_(-0.40)) × 10^(-3), where the errors refer to the experimental and theoretical uncertainties. From a simultaneous fit to the data over the full q^2 range and the FNAL/MILC lattice QCD results, we obtain |V_(ub)|=(2.95 ± 0.31) × 10^(-3) from B → πℓν, where the error is the combined experimental and theoretical uncertainty
