In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of $λ$-Hölder continuous functions on $[-τ,0]$, for some suitable $τ>0$ and $λ\in(1/2,H)$. The rate of convergence of our scheme is $1/n^γ$, for any $γ<2λ-1$. Also, numerical simulations are provided to illustrate our theoretical results.