We report on a Dalitz plot analysis of B^-→D^+π^-π^- decays, based on a sample of about 383×10^6 Υ(4S)→BB̅ decays collected with the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. We find the total branching fraction of the three-body decay: B(B^-→D^+π^-π^-)=(1.08±0.03±0.05)×10^(-3). We observe the established D_2^(*0) and confirm the existence of D_0^(*0) in their decays to D^+π^-, where the D_2^(*0) and D_0^(*0) are the 2^+ and 0^+cu̅ P-wave states, respectively. We measure the masses and widths of D_2^(*0) and D_0^(*0) to be: m_(D2)^(*0)=(2460.4±1.2±1.2±1.9) MeV/c^2, Γ_(D2)^(*0)=(41.8±2.5±2.1±2.0) MeV, m_(D0)^(*0)=(2297±8±5±19) MeV/c^2, and Γ_(D0)^(*0)=(273±12±17±45) MeV. The stated errors reflect the statistical and systematic uncertainties, and the uncertainty related to the assumed composition of signal events and the theoretical model.