Vaidhyanathan is the city's first Indian-American mayor and takes over from Barry Chang.
"This is definitely a very momentous moment in my life," Vaidhyanathan said during the hearing.
"I have several people to thank for this, and many of them are in this room, and my father, who is not here but is watching from heaven above. He instilled in me the confidence that I could do anything I set my heart to as long as I put the hard work in," she added.
Vaidhyanathan was first elected to the city council in November 2014 and will begin serving as mayor in 2017.
She thanked her, mother, who had come to the ceremony from India, and her daughter, husband and other family members in attendance.
"I've had several congratulatory messages saying that I'm the first woman mayor of Indian origin," she said. "Yes, I do take a lot of pride and prestige in that, but I do want to thank the residents of the city of Cupertino that voted me in not looking at ethnicity at all. Thank you for your trust in me, and maybe we did break that silicon ceiling and put a few more cracks in that glass ceiling."
She mentioned that she is looking forward to continuing to work with residents in the community.
Vaidhyanathan is also the first woman mayor of the city since Kris Wang served in 2010.