Elastic properties and phase stabilities of L12-A13Sc precipitate phase in Al-Sc alloys have been topics of experimental and theoretical research over the past years. However, these properties of off-stoichiometric L12-A13Sc have not been investigated. Firstly, in combination with Wagner-Schottky model, the first-principles total energy calculations based on density functional theory are performed to study point defect concentrations of intermetallic L12-A13Sc each as a function of temperature and alloy composition. We calculate the point defect formation enthalpies and plot the point defect density curves of stoichiometric and off-stoichiometric L12-A13Sc at 1000 K. The results show that within the whole temperature range (300-1200 K), Al and Sc vacancies dominate on stoichiometric L12-A13Sc but with low concentrations (~10-6 even at 1200 K); on the Al-rich side of off-stoichiometric L12-A13Sc, the Al anti-site and the Sc vacancy concentrations dominate, and their concentrations are comparable, however, on Sc-rich side of off-stoichiometric L12-A13Sc, the Sc anti-site defect dominates. Furthermore, the lattice constants and the elastic constants of stoichiometric and off-stoichiometric L12-A13Sc are calculated, and it is worth noting that 222 supercell models with a point defect are used for off-stoichiometric L12-A13Sc in the calculation. Then employing calculated elastic constants, the values of Youngs modulus, shear modulus, bulk modulus, anisotropic index, G/B ratio, Cauchy pressure, and Poisson ratio of stoichiometric and off-stoichiometric L12-A13Sc are computed. And lastly, combining these data with point defect concentrations of off-stoichiometric L12-A13Sc at 1000 K, the comprehensive effects of four point defects on elastic properties of L12-A13Sc are evaluated. The four point defects coexist in L12-A13Sc as we know from the calculations of equilibrium point defect density. The conclusions are as follows. 1) The point defects can cause off-stoichiometric L12-A13Sc lattice distortion. On the Sc-rich side, lattice constant appears to be an increasing tendency, from 4.105 to the biggest value of ~4.13 (~0.5% growth), while on the Al-rich side, it shows an opposite trend, from 4.105 to the smallest value of ~4.10 (~0.24% fall). Although there is the lattice distortion in off-stoichiometric L12-A13Sc, off-stoichiometric L12-A13Sc can still keep stable crystal structure for the value of xAl in a range of 0.72-0.78. 2) The point defects also affect elastic constants of off-stoichiometric L12-A13Sc. Specifically, on the Sc-rich side, elastic constant c11 decreases with the increase of deviation degree of stoichiometric ratio, and the maximal reduction is ~9% at xAl = 0.72, while elastic constants c12 and c44 show the opposite variation trends, and the maximal increase is ~8% at xAl = 0.72. On the Al-rich side, there are little changes for elastic constants c11, c12 and c44. 3) The point defects obviously increase the elastic anisotropy of off-stoichiometric L12-A13Sc, and especially on the Sc-rich side, the notable increase is found, which jumps from 1.610-6 to 0.04. 4) The values of Youngs modulus, shear modulus, and bulk modulus of off-stoichiometric L12-A13Sc decrease due to point defects, with the maximal reduction being 3%-4%. These elastic modules fall first rapidly and then slowly on the Sc-rich side, while they present approximately a linear downward trend on the Al-rich side. In addition, weak influences are exerted on brittleness and toughness of off-stoichiometric L12-A13Sc by the point defects, compared with the other elastic effects mentioned above. In summary, in the scope of xAl = 0.72-0.78, the point defects can not only reduce Youngs modulus, shear modulus, and bulk modulus of off-stoichiometric L12-A13Sc, but also increase the anisotropies of the elastic properties of off-stoichiometric L12-A13Sc. However, the point defects have weak influences on the brittleness and toughness of off-stoichiometric L12-A13Sc.