You have to join the Mages College. The first quest they give you will send you off to Saathal.
This. I'm assuming you're doing the Forbidden Legend quest? I just finished doing that quest chain myself. The reward is totally worth the inconvenience of having to join the College of Winterhold. I'd imagine it's a mistake on Bethesda's part to not make Saarthal accessible without College membership though, it might be patched.