Carbon dioxide (CO2) sequestration in saline aquifers is considered as one of the most viable and promising ways to reduce CO2 concentration in the atmosphere. CO2 is injected into deep saline formations at supercritical state where its density is smaller than the hosting brine. This motivates an upward motion and eventually CO2 is trapped beneath the cap rock. The trapped CO2 slowly dissolves into the brine causing the density of the mixture to become larger than the host brine. This causes gravitational instabilities that is propagated and magnified with time. In this kind of density-driven flows, the CO2-rich brines migrate downward while the brines with low CO2 concentration move upward. With respect to the properties of the subsurface aquifers, there are instances where saline formations can possess anisotropy with respect to their hydraulic properties. Such anisotropy can have significant effect on the onset and propagation of flow instabilities. Anisotropy is predicted to be more influential in dictating the direction of the convective flow. To account for permeability anisotropy, the method of multipoint flux approximation (MPFA) in the framework of finite differences schemes is used. The MPFA method requires more point stencil than the traditional two-point flux approximation (TPFA). For example, calculation of one flux component requires 6-point stencil and 18-point stencil in 2-D and 3-D cases, respectively. As consequence, the matrix of coefficient for obtaining the pressure fields will be quite complex. Therefore, we combine the MPFA method with the experimenting pressure field technique in which the problem is reduced to solving multitude of local problems and the global matrix of coefficients is constructed automatically, which significantly reduces the complexity. We present several numerical scenarios of density-driven flow simulation in homogeneous, layered, and heterogeneous anisotropic porous media. The numerical results emphasize the significant effects of anisotropy in driving the migration of dissolved CO2 along the principal direction of anisotropy even if the porous medium is highly heterogeneous. Furthermore, the impacts of the increase of density difference between the brine and the CO2-saturated brine with respect to the onset time of convection, the CO2 flux, and the CO2 total dissolved mass are also discussed in this paper.