Morton, 47, will replace 65-year-old Chan Gailey, who held the job for two years before announcing his retirement after the 2016 season.
The Jets finished 26th in total offense and 30th in scoring this season.
Morton, a 13-year NFL coach, spent the past two seasons with the Saints as receivers coach. He held the same position with the San Francisco 49ers under Jim Harbaugh from 2011-2014, and the 49ers went to three NFC Championship Games and one Super Bowl in that time.
Morton first joined the Saints in 2006 as an offensive assistant before returning to New Orleans in 2015. He broke into the NFL coaching ranks in 2002 as an offensive assistant with the Oakland Raiders.
Morton worked at the University of Southern California as a wide receivers coach and passing game coordinator with two seasons as offensive coordinator in 2009-10.