Intersubband transitions stimulated by an electromagnetic wave which has a momentum in the plane of a two-dimensional electron gas (2DEG) are accompanied by a ``photon-drag'' current due to the momentum imparted to electrons by the absorbed photons. In a high-mobility 2DEG at low temperatures, an enhanced effect occurs owing to the difference in the momentum relaxation times in the ground and the excited subbands. A theory of this effect is developed with particular attention paid to the geometrical aspects of proposed experiments. The enhanced effect is considered in two limits, corresponding to two dominant mechanisms of line broadening in the intersubband absorption spectra (the collision and the Doppler linewidths). The photon-drag effect permits a new type of spectroscopy containing information about the momentum-relaxation kinetics in 2D subbands. Moreover, it allows the implementation of novel far-infrared detectors. Fundamental limits on the quantum efficiency of such detectors are discussed.