The 53-year-old, who resigned as the RFU's director of professional rugby in April, succeeds Zac Toumazi.
He will begin his new role at Hove in January 2017.
"He has exceptional experience, both playing and administering sport, which will be of great help to our cricket management delivering success," said Sussex chairman Jim May.
"Rob will be joining us at an important moment and he fully believes in our strategy of increasing the quantity and quality of participation in the recreational game."
Andrew made his England debut in 1985 against Romania, and become a key member of the side that won three Grand Slams and reached the 1991 World Cup final.
After helping England to the semi-finals of the 1995 World Cup, he moved from Wasps to Newcastle, where he won the Premiership title at the first attempt in 1998.
After 10 years with the Falcons he left to join the RFU as elite rugby director, a role he held for five years before becoming rugby operations director and director of professional rugby.
He was also an accomplished cricketer, winning two Blues while at Cambridge University and scoring a first-class hundred against Nottinghamshire in 1984.
He also made a handful of appearances for Yorkshire's second team.
"I am looking forward to working with everyone connected with Sussex to help create success in professional cricket, recreational cricket and community programmes and ensuring that Hove remains a very special cricket ground," said Andrew.
"I will be focussing on driving forward the strategy over the next few years and helping write the next chapter in the rich history of cricket in Sussex.
"These are exciting times for Sussex cricket and cricket in general, and I can't wait to get started."
Sussex have appointed former Rugby Football Union director and England fly-half Rob Andrew as chief executive.