Finite integration method is employed to solve convection-diffusion equations. For diffusion-dominated convection-diffusion problems, numerical results show that finite integration method is more than one and two orders of magnitude better in accuracy than QUICK scheme and central differencing scheme, respectively. We find that finite integration method can be improved by introducing a weighting parameter in numerical quadrature. By appropriately adjusting the parameter, modified finite integration method is at least more than four orders of magnitude better in accuracy than traditional finite volume method for convection-dominated flows. Furthermore, even coarse grids are utilized to divide flow domain modified finite integration method has not lead to unphysical oscillations in convection-dominated convection-diffusion problem, which demonstrates a good stability of the method.