We report the first detection of the ground-state rotational transition of the methylidyne cation CH+ towards the massive star-forming region DR21 with the HIFI instrument onboard the Herschel satellite. The line profile exhibits a broad emission line, in addition to two deep and broad absorption features associated with the DR21 molecular ridge and foreground gas. These observations allow us to determine a CH+ J=1-0 line frequency of 835137 +/- 3 MHz, in good agreement with a recent experimental determination. We estimate the CH+ column density to be a few 1e13 cm^-2 in the gas seen in emission, and > 1e14 cm^-2 in the components responsible for the absorption, which is indicative of a high line of sight average abundance [CH+]/[H] > 1.2x10^-8. We show that the CH+ column densities agree well with the predictions of state-of-the-art C-shock models in dense UV-illuminated gas for the emission line, and with those of turbulent dissipation models in diffuse gas for the absorption lines.