Exclusive electroproduction of mesons was studied by scattering 27.6 GeV positrons or electrons off a transversely polarized hydrogen target. The single-spin azimuthal asymmetry with respect to target polarization was measured as a function of the Mandelstam variable t, the Bjorken scaling variable , and the virtuality of the exchanged photon. The extracted Fourier components of the asymmetry were found to be consistent with zero, except one that was found to be large and that involves interference of contributions from longitudinal and transverse virtual photons.