A fourth-order compact finite-difference method is proposed in this paper to solve the system of two-dimensional Burgers' equations. The new method is based on the two-dimensional Hopf–Cole trans-formation, which transforms the system of two-dimensional Burgers' equations into a linear heat equation. The linear heat equation is then solved by an implicit fourth-order compact finite-difference scheme. A compact fourth-order formula is also developed to approximate the boundary conditions of the heat equation, while the initial condition for the heat equation is approximated using Simpson's rule to maintain the overall fourth-order accuracy. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. Copyright © 2009 John Wiley & Sons, Ltd.