Actually, it does teach that grace saves and that we cannot be saved in any other way no matter what we do.Actually, Mormonism doesn't teach that grace saves.
If it was by works alone, then grace would be nullified. Since that isn't what we teach then grace is still necessary and therefore not nullified. That's just your twist.It's grace + works, which nullifies the grace.
Try this twist on for size. If it was by grace alone without works, then why isn't everyone saved?