Using a graph counting technique suitable for regular fractals, an exact evaluation of the total number of embeddings of self-avoiding walks on the generalized Sierpinski gasket is obtained. Numerical estimates for the connective constants are quoted for the first time, where b is the generation parameter of the gaskets. It is shown that the number of distinct n-step SAWs per site converges to the triangular lattice values when . Our analysis indicates that converges to the Euclidean value in the same limit and an asymptotic expression is given.