Detecting neutral hydrogen (H i) via the 21 cm line emission from the intergalactic medium at z≳6 has been identified as one of the most promising probes of the epoch of cosmic reionization-a major phase transition of the Universe. However, these studies face severe challenges imposed by the bright foreground emission from cosmic objects. Current techniques require precise instrumental calibration to separate the weak H i line signal from the foreground continuum emission. We propose to mitigate this calibration requirement by using measurements of the interferometric bispectrum phase. The bispectrum phase is unaffected by antenna-based direction-independent calibration errors and hence for a compact array it depends on the sky brightness distribution only (subject to the usual thermal-like noise). We show that the bispectrum phase of the foreground synchrotron continuum has a characteristically smooth spectrum relative to the cosmological line signal. The two can be separated effectively by exploiting this spectral difference using Fourier techniques, while eliminating the need for precise antenna-based calibration of phases introduced by the instrument, and the ionosphere, inherent in existing approaches. Using fiducial models for continuum foregrounds, and for the cosmological H i signal, we show the latter should be detectable in bispectrum phase spectra, with reasonable significance at |k_{∥}|≳0.5h  Mpc^{-1}, using existing instruments. Our approach will also benefit other H i intensity mapping experiments that face similar challenges, such as those measuring baryon acoustic oscillations (BAO).