We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝdefined on a connected, second countable Finsler manifoldM, for each positive continuous functionε:M→(0,∞)and eachr>0, there exists aC1-smooth Lipschitz functiong:M→ℝsuch that|f(x)-g(x)|≤ε(x), for everyx∈M, andLip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebraCb1(M)of allC1functions with bounded derivative on a complete quasi-reversible Finsler manifoldM, we obtain a characterization of algebra isomorphismsT:Cb1(N)→Cb1(M)as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.