Acoustic solitons formed during the propagation of a picosecond strain pulse in a GaAs crystal with a ZnSe/ZnMgSSe quantum well on top lead to exciton resonance energy shifts of up to 10 meV, and ultrafast frequency modulation, i.e., chirping, of the exciton transition. The effects are well described by a theoretical analysis based on the Korteweg-de Vries equation and accounting for the properties of the excitons in the quantum well.