Context. Several exoplanet direct-imaging instruments (VLT-SPHERE, Gemini Planet Imager, etc.) will soon be in operation, providing original data for comparative exoplanetary science to the community. To this end, exoplanet imagers use an extreme adaptive optics (XAO) system to correct the atmospheric turbulence and provide a highly corrected beam to a near-infrared (NIR) coronagraph for suppressing diffracted stellar light. The performance of the coronagraph is, however, limited by the non-common path aberrations (NCPA) due to the differential wavefront errors existing between the visible XAO sensing path and the NIR science path and leading to residual speckles that hide the faintest exoplanets in the coronagraphic image. Aims. Accurate calibration of the NCPA in exoplanet imagers is mandatory to correct the residual, quasi-static speckles remaining in the coronagraphic images after XAO correction in order to allow the observation of exoplanets that are at least 106 fainter than their host star. Several approaches have been developed during these past few years to reach this goal. We propose an approach based on the Zernike phase-contrast method operating in the same wavelength as the coronagraph for the measurements of the NCPA between the optical path seen by the visible XAO wavefront sensor and that seen by the NIR coronagraph. Methods. This approach uses a focal plane phase mask of size ~λ/D, where λ and D denote the wavelength and the telescope aperture diameter, respectively, to measure the quasi-static aberrations in the upstream pupil plane by encoding them into intensity variations in the downstream pupil image. The principle of this approach as described in several classical optical textbooks is simplified by the omission of the spatial variability of the amplitude diffracted by the phase mask. We develop a more rigorous formalism, leading to highly accurate measurement of the NCPA, in a quasi-linear way during the observation. Results. With prospects of achieving subnanometric measurement accuracy with this approach for a static phase map of standard deviation 44 nm rms at λ = 1.625   μm (0.026 λ), we estimate a possible reduction of the NCPA due to chromatic differential optics by a factor ranging from 3 to 10 in the presence of adaptive optics (AO) residuals compared with the expected performance of a typical current-generation system. This would allow a reduction of the level of quasi-static speckles in the detected images by a factor 10 to 100, thus correspondingly improving the capacity to observe exoplanets.