We investigate neutrino oscillations from core-collapse supernovae that produce magnetohydrodynamic (MHD) explosions. By calculating numerically the flavor conversion of neutrinos in the highly non-spherical envelope, we study how the explosion anisotropy has impacts on the emergent neutrino spectra through the Mikheyev-Smirnov-Wolfenstein effect. In the case of the inverted mass hierarchy with a relatively large theta(13) (sin(2) 2 theta(13) greater than or similar to 10(-3)), we show that survival probabilities of (nu) over bar (e) and nu(e) seen from the rotational axis of the MHD supernovae (i.e., polar direction), can be significantly different from those along the equatorial direction. The event numbers of (nu) over bar (e) observed from the polar direction are predicted to show steepest decrease, reflecting the passage of the magneto-driven shock to the so-called high-resonance regions. Furthermore we point out that such a shock effect, depending on the original neutrino spectra, appears also for the low-resonance regions, which could lead to a noticeable decrease in the nu(e) signals. This reflects a unique nature of the magnetic explosion featuring a very early shock-arrival to the resonance regions, which is in sharp contrast to the neutrino-driven delayed supernova models. Our results suggest that the two features in the (nu) over bar (e) and nu(e) signals, if visible to the Super-Kamiokande for a Galactic supernova, could mark an observational signature of the magnetically driven explosions, presumably linked to the formation of magnetars and/or long-duration gamma-ray bursts.