Recently, numerical solutions to the field equations of Einstein-scalar-Gauss-Bonnet (EsGB) gravity that correspond to black holes with nontrivial scalar hair have been reported. Here, we employ the method of the continued-faction expansion in terms of a compact coordinate in order to obtain an analytical approximation for the aforementioned solutions. For a wide variety of coupling functionals to the Gauss-Bonnet term we were able to obtain analytical expressions for the metric functions and the scalar field. In addition we estimated the accuracy of these approximations by calculating the black-hole shadows for such black holes. Excellent agreement between the numerical solutions and analytical approximations has been found.