Vanessa Hudgens is officially a mother ... she gave birth to her first child, and it looks like things went well, because she's taking her bundle of joy home.
Check out these photos of Vanessa leaving the hospital Wednesday in Santa Monica ... where a nurse wheeled out to her car with the baby swaddled in her arms.
You don't see him in the pic, but the proud father, Vanessa's MLB player husband Cole Tucker was also there .