Auger decay carries valuable information about the electronic structure and dynamics of atoms, molecules, and solids. Here we furnish evidence that under certain conditions Auger electrons are subject to an energetic chirp. The effect is disclosed in time-resolved streaking experiments on the Xe NOO and Kr MNN Auger decay using extreme-ultraviolet pulses from the free-electron laser in Hamburg as well as from a high-order harmonic laser source. The origin of this effect is found to be an exchange of energy between the Auger electron and an earlier emitted correlated photoelectron. The observed time-dependent spectral modulations are understood within an analytical model and confirmed by extensive computer simulations.