Gangwal - who co-founded InterGlobe Aviation (INGL.NS), the operator of India's largest airline IndiGo - has also served on the boards of US Airways Group, CarMax (KMX.N) and Office Depot, among others.
"Rakesh's expertise in travel technology will be valuable as we continue to make investments that support our operations and strategic initiatives," said Gary Kelly, executive chairman of Southwest's board.
Elliott, known for pushing for a change to boost shareholder returns, had called for a reconstitution of Southwest's board with "new, truly independent" directors possessing expertise in airlines, customer experience and technology.
The activist investor had also called for a leadership change citing Southwest's disappointing financial results.
Last week, Southwest adopted a "poison pill", used by corporate boards to thwart hostile takeover bids, to deter Elliott Investment Management from raising its stake in the carrier.
"Among the criteria for selecting this new director was clearly that he would be supportive of Southwest's current leadership and status-quo approach," Elliott said in a letter to Southwest's board on Monday.
Elliott currently owns an 11% stake in the company.
