We report the discovery of low-amplitude gravity-mode oscillations in the massive binary star V380 Cyg, from 180 d of Kepler custom-aperture space photometry and 5 months of high-resolution high signal-to-noise ratio spectroscopy. The new data are of unprecedented quality and allowed the improvement of the orbital and fundamental parameters for this binary. The orbital solution was subtracted from the photometric data and led to the detection of periodic intrinsic variability with frequencies, of which some are multiples of the orbital frequency and others are not. Spectral disentangling allowed the detection of line-profile variability in the primary. With our discovery of intrinsic variability interpreted as gravity-mode oscillations, V380 Cyg becomes an important laboratory for future seismic tuning of the near-core physics in massive B-type stars.