The elastic properties and solid-solution strengthening (SSS) of the binary Ni–Co and Ni–Cr, and ternary Ni–Co–Cr alloys were investigated by the first-principles method. The results show that both Co and Cr increase lattice parameters of the binary alloys linearly. However, nonlinearity is found in compositional dependence of lattice parameters in the ternary Ni–Co–Cr alloys, that is, Co increases but decreases the lattice parameter at low and high Cr concentrations, respectively. Co increases the bulk, shear, and Young’s moduli (B, G, and E), while Cr increases B but decreases G and E in the binary alloys. In the ternary Ni–Co–Cr alloys, G and E have a similar compositional dependence to those in the binary alloys, except for B. Based on the Labusch model, the SSS parameter of Ni–Cr is larger than that of Ni–Co. The SSS effect increases significantly with Cr addition, especially at low Co concentrations in the ternary Ni–Co–Cr alloys. Meanwhile, it increases mildly with Co addition at low Cr concentrations but decreases with Co addition at high Cr concentrations.