Shah, who has worked as a consultant for the Cabinet Office and HSBC, joins the Robins board which is led by chairman Jed McCrory with a view to putting the club on a stable financial footing.
"I have utter delight at the opportunity to serve the club," she said. "The vision and strategy that Jed has for the club is compelling and one that will result in Swindon Town flourishing.
"Already, it is reaching a stable financial footing and with the further changes planned, I am confident that the board will move the club rapidly to a position where it will be thriving - both on and off the pitch."
McCrory added: "I am very pleased that a person of Sangita's gravitas and business acumen has agreed to join us in our push to develop a community club here at Swindon Town.
"I have worked with Sangita over the past five or six years on numerous different sector projects and I have no doubt that she will bring expertise that will help us continue the work we have started."