Faint vibration-rotation quadrupole emission of H2 has been detected in the infrared spectrum of a bright optical shock-excited emission-line filament to the NE of the Cygnus Loop supernova remnant. The optical line emission is excited in fast shocks (100 km s-1) in low density gas (10 cm-3). These are novel conditions for the excitation of H2. The H2 1-0 S(1) emission at 2.122-mu-m has been imaged and filamentary emission is found to extend over a region of at least 7' X 7'. One ridge of emission lies 2' beyond the edge of the remnant as delineated by the optically bright radiative shocks, and is apparently associated with the faint "nonradiative" Balmer-dominated shocks in this vicinity. However, there are several locations where H2 and bright optical line emission are coincident. A spectrum of an H2 filament discovered in the infrared image shows H2 1-0 S(2), 1-0 S(1), 1-0 S(0), and 2-1 S(1). The relative intensity of these lines can be used to show that the H2 level population is consistent with a single rotational and vibrational excitation temperature of 2200 +/- 500 K. This result rules out UV fluorescence and molecule formation schemes as the emission mechanism. To explain these results we invoke a shock which is intermediate between a simple hydrodynamic J-shock and a magnetohydrodynamic C-shock where there is no shock discontinuity-a J shock with a magnetic precursor. In such a shock the ion-magnetosonic speed is greater than the shock velocity and magnetosonic waves propagate ahead of the shock. As these waves are damped the magnetic field is compressed, forcing ions and electrons ahead of the shock to stream through the neutral gas, heating the preshock gas. We present calculations which show that excitation of H2 in a magnetic precursor to a J-shock explains quantitatively the observed H2 surface brightness, level population, and relation to optical emission. The presence of a magnetic shock precursor may also help us to understand some of the anomalous properties of nonradiative shocks.