We present an efficient method to obtain single-crystal elastic constants. Using the method, the computation effort for the complete set of elastic constants (21 components for a triclinic crystal) of a crystal under a given strain state is similar to that for the bulk modulus, where the latter is normally calculated first together with the equilibrium volume by fitting the volume-dependent energy to an equation of state. The idea in the efficiency improvement is to apply linear-independent coupling strains to a crystal, and to extract the complete set of elastic constants simultaneously. For the new method, a single set of strains are universally applicable to all crystals, and less number of strains are needed for crystals of higher symmetry. Calculated single-crystal elastic constants of cubic diamond, hexagonal osmium, orthorhombic TiSi2 and monoclinic Mg5Si6 are given as examples.