We present Very Large Array (VLA) observations of 21 cm H I absorption in the high-velocity (HV) system toward the external galaxy NGC 1275 (Per A). The primary motivation for our observations is to examine whether the only existing (single-dish) detection of the magnetic field in an external galaxy will stand up to independent scrutiny with an interferometer. The H I absorption profile in the HV system toward NGC 1275 has a number of velocity components. In this paper, we focus on the prominent narrow component at 8114 km s^-1, in which we have detected the Zeeman effect. The measured line-of-sight magnetic field is B_los = -21.5 ± 6.5 μG. Within the errors, this is consistent with the single-dish detection. A weighted average of the single-dish and VLA detections gives a value for the line-of-sight magnetic field of B_los = -10.7 ± 3.2 μG. The derived ratio of total internal kinetic energy to gravitational energy suggests that the clouds responsible for H I absorption are likely not gravitationally bound. The ratio of magnetic energy to total internal kinetic energy indicates that magnetic effects are likely dominant in these clouds. Finally, the measured ratio of mass to magnetic flux is found to be much lower than the critical value, indicating that the observed H I absorbing clouds are significantly subcritical. This agrees with a recent study of H I absorbing clouds in the cold neutral medium of our Galaxy, thereby extending an important result for Galactic clouds to an extragalactic case.