We show theoretically that an electric current in a high-mobility quasi-two-dimensional electron layer induces a significant drift of excitons in an adjacent layer through interlayer Coulomb interaction. The exciton gas is shown to drift with a velocity which can be a significant fraction of the electron drift velocity at low temperatures. The estimated drift length is of the order of micrometers or larger during the typical exciton lifetime for GaAs/AlxGa1-xAs double quantum wells. A possible enhancement of the exciton radiative lifetime due to the drift is discussed.