The definition of an exponential B-spline function and its basic derivative formats on a rectangular uniform grid were given. For a given Burgers equation, the exponential B-spline scheme was used for discretization in the space direction and the Crank-Nicolson scheme combined with the Hermite compact scheme was used for discretization in the time direction. The numerical solution was obtained, and the stability and convergence of this scheme were proved. The errors and convergence orders of the numerical solution and the influence of different parameter values on the precision of the numerical solution were studied experimentally.