An efficient numerical algorithm is proposed for the Ericksen-Leslie equations with variable density. Firstly, by introducing a SAV(scalar auxiliary variable) with the free energy, an equivalent new system is obtained. Secondly, a numerical scheme of the new system is established where the Ginzburg-Landau penalty function is handled explicitly such that the nonlinear terms are linearized. Theoretical analysis has demonstrated the unique solvability and unconditional energy stability of the scheme, and the first-order convergence rate of the scheme has been proved by rigorous error analysis. The theoretical derivation results were verified by several numerical simulations, and the singularity annihilation process was presented. The constructed scheme maintains the expected accuracy in both theoretical and numerical calculations, and exhibits good performance in evolutionary simulations.