Beam-target double-spin asymmetries and target single-spin asymmetries were measured for the exclusive π+ electroproduction reaction γ∗p→nπ+. The results were obtained from scattering of 6 GeV longitudinally polarized electrons off longitudinally polarized protons using the CEBAF Large Acceptance Spectrometer at Jefferson Lab. The kinematic range covered is 1.1<W<3 GeV and 1<Q2<6 GeV2. Results were obtained for about 6000 bins in W, Q2, cos(θ∗), and ϕ∗. Except at forward angles, very large target-spin asymmetries are observed over the entire W region. Reasonable agreement is found with phenomenological fits to previous data for W<1.6 GeV, but very large differences are seen at higher values of W. A GPD-based model is in poor agreement with the data. When combined with cross section measurements, the present results provide powerful constraints on nucleon resonance amplitudes at moderate and large values of Q2, for resonances with masses as high as 2.4 GeV.