The effect of copper alloying up to 25 at % on the structure–phase transformations and the physicomechanical properties of ternary alloys from the quasi-binary TiNi–TiCu section is studied by measuring the physicomechanical properties, transmission electron microscopy, scanning electron microscopy, electron diffraction, and X-ray diffraction (XRD). The data of temperature measurements of the electrical resistivity and the magnetic susceptibility and XRD data are used to plot a general diagram for the thermoelastic B2 ↔ B19', B2 ↔ B19 ↔ B19', and B2 ↔ B19 martensitic transformations, which occur in the alloys upon cooling as the copper content increases in the ranges 0–8, 8–15, and 15–25 at % Cu, respectively. The experimental results are compared to the well-known data, including differential scanning calorimetry data, obtained for these alloys. The changes in the mechanical properties and the microstructure of the alloys in the state of B19 or B19' martensite are discussed.