We present measurements of the complete set of third-order elastic constants (TOEC) of Cu-Al-Ni and Cu-Al-Be shape-memory alloys, as functions of temperature. These data quantify the elastic anharmonic behavior of these alloys on approaching their martensitic transformation. We provide experimental evidence that the martensitic transition in Cu-based shape-memory alloys occurs at a fixed value (~=60 GPa) of the TOEC combination C3=(1/83)(3C112-2C123-C111), which is the relevant combination for the shear mechanisms leading the bcc structure to the close-packed martensite. By means of a Landau model for elastic phase transitions, the value of C3 has been estimated from the crystal structure of the martensitic phase. The value obtained conforms reasonably with that found experimentally.