Infrared spectroscopy of V2274 Cygni (=Nova Cygni 2001 No. 1) is presented for two widely separated epochs, 17 and 370 days after discovery. In addition to the Paschen and Brackett series of H I, the early-time spectrum shows strong emission lines of C I and N I, fluorescently excited lines of O I, and emission from the first overtone of carbon monoxide. Because the initial data were probably acquired no more than 18 days after outburst, CO molecule formation occurred remarkably quickly in the dense, cool, carbon and oxygen rich ejecta. Rapid formation was also seen in NQ Vul, V842 Cen, and V705 Cas, three other novae in which first-overtone CO emission has been detected. Formation of the CO molecule may occur chemically in a process that requires H2 as a precursor or directly through radiative association. The overtone emission of V2274 Cyg indicates a temperature of ~2500 K. The vibrational levels show no obvious departures from thermal equilibrium, which may indicate high optical depths in the fundamental. A large 13C/12C ratio (0.83 ± 0.3) is also indicated by the observations, consistent with the fast CNO burning expected in novae explosions. By the time of the second epoch observations, the emission lines of the neutral C, N, and O had disappeared. He I λ10830 was the dominant emission feature in the spectrum. In addition to the hydrogen lines, recombination features of He II were also strong. The common nebular lines of [S III] were seen but only two coronal line, [S VIII] λ9911 and [Si VI] λ19645, were detected. The CO emission had disappeared, but a strong thermal dust component was present. The interstellar reddening for the system was found to be E(B-V) = 1.3. This extinction, together with the absolute magnitude derived from the rate of decline of the light curve, suggest a distance of ~10.8 kpc. This places V2274 Cyg well out of the Galactic plane. The small number of novae with spectroscopic detections of carbon monoxide all have prominent C I lines, moderate speed classes and ejection velocities, exhibit marked dust formation events, and result from an explosion on a CO-type white dwarf. Based on these similarities, the spectrum of V2274 is proposed as a likely near-infrared spectral template for other novae that display carbon monoxide emission.