The finite-difference based eikonal solution for a point-source initial condition has an upwind singularity at the source position. As a result, even the high-order finite-difference schemes exhibit at most polluted first-order convergence, as the errors around the source location spread to the whole computational domain. We apply factorization approach to obtain clean first-order solutions for the tilted transversely isotropic (TTI) eikonal equation. The idea relies on factoring the unknown traveltime function into two terms. One of these two factors is utilized to capture the source singularity; thus, the other function is smooth in the neighborhood of the source. We solve a sequence of factored tilted elliptically anisotropic eikonal equations, each time by updating the effective velocities needed to capture the higher order nonlinearity of the TTI eikonal equation. We design an iterative monotone fast sweeping scheme to compute the TTI eikonal solution. Numerical tests show significant improvement in accuracy due to the factorization approach compared to directly solving the eikonal equation.