Beam-target double-spin asymmetries and target single-spin asymmetries were measured for the exclusive
π
+
electroproduction reaction
γ
∗
p
→
n
π
+
. The results were obtained from scattering of 6-GeV longitudinally polarized electrons off longitudinally polarized protons using the CEBAF Large Acceptance Spectrometer at Jefferson Laboratory. The kinematic range covered is
1.1
<
W
<
3
GeV and
1
<
Q
2
<
6
GeV
2
. Results were obtained for about 6000 bins in
W
,
Q
2
,
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 generalized parton distributions (GPD)-based model is in poor agreement with the data. When combined with cross-sectional measurements, the present results provide powerful constraints on nucleon resonance amplitudes at moderate and large values of
Q
2
, for resonances with masses as high as 2.4 GeV