We present results of a high angular resolution survey of massive OB stars in the Cygnus OB2 association that we conducted with the Near-Infrared Imager camera and ALTAIR adaptive optics system of the Gemini North telescope. We observed 74 O- and early-B-type stars in Cyg OB2 in the JHK infrared bands in order to detect binary and multiple companions. The observations are sensitive to equal-brightness pairs at separations as small as 0.08", and progressively fainter companions are detectable out to {Delta}K=9mag at a separation of 2". This faint contrast limit due to read noise continues out to 10" near the edge of the detector. We assigned a simple probability of chance alignment to each companion based upon its separation and magnitude difference from the central target star and upon areal star counts for the general star field of Cyg OB2. Companion stars with a field membership probability of less than 1% are assumed to be physical companions. This assessment indicates that 47% of the targets have at least one resolved companion that is probably gravitationally bound. Including known spectroscopic binaries, our sample includes 27 binary, 12 triple, and 9 systems with 4 or more components. These results confirm studies of high-mass stars in other environments that find that massive stars are born with a high-multiplicity fraction. The results are important for the placement of the stars in the Hertzsprung-Russell diagram, the interpretation of their spectroscopic analyses, and for future mass determinations through measurement of orbital motion.