The authors propose an algorithm to construct Chebyshev approximation for functions of several variables by a generalized polynomial as a limiting approximation in the norm of space L p as p → ∞. It is based on serial construction of power-average approximations using the least squares method with variable weight function. The convergence of the method provides an original way to consistently refine the values of the weight function, which takes into account the results of approximation at all previous iterations. The authors describe the methods of calculating the Chebyshev approximation with absolute and relative errors. The results of test examples confirm the efficiency of using the method to obtain Chebyshev approximation of tabular continuous functions of one, two, and three variables.