The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix rho . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by N massless Dirac fermions. For the density matrix, rho _A, of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of rho _A; we derive mathrm {tr}(mathrm {e}^{mathrm {i}alpha Q_A} rho _A^n)~=~a mathrm {e}^{mathrm {i}alpha langle Q_Arangle } (sigma L)^{-x}(1+O(L^{-mu })), where a, x and mu are universal and sigma depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term O(L^{-mu }) is consistent with but weaker than the field theory prediction O(L^{-2mu }). However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.