The classical symmetries are expanded by calculating the potential symmetries of nonlinear telegraph-type (NTT) and Burgers equations, and a series of new exact solutions of Burgers equation are obtained. The classical symmetries and potential symmetries of NTT and Burgers equations are determined based on differential characteristic set algorithm, and one-parameter Lie transformation group of two potential symmetries to Burgers equation is calculated. The invariant solutions of Burgers equation are constructed by using generalized simple equation method, and these solutions are expressed by hyperbolic functions, trigonometric functions and rational functions with any two parameters respectively. New exact solutions are obtained by acting Lie transformation group (14) of potential symmetry on Burgers equation, and these solutions can’t be obtained by the classical symmetry of the equation.