Gordon, 45, succeeds Ted Nolan, who was fired in July.
An Islanders announcement said Gordon received a multiyear contract.
"Scott has been one of the top up-and-coming coaches in hockey and with good reason," said Garth Snow, the Islanders' general manager. "Through his work in the minor leagues and, most recently, with Providence ... he has gained an impeccable reputation. The entire Islanders organization looks forward to working with Scott."
Gordon had a 221-141-20-27 record with Providence.
"Garth has given me an unbelievable opportunity to coach in the NHL with the Islanders," Gordon said.
"From my time coaching in Providence, I've seen firsthand the talented young players in this organization. I look at the direction and the vision for this team and see a bright future."
Gordon, of Brockton, Mass., began his minor-league coaching career in 1994 with the the Atlanta Knights in the International Hockey League.
Nolan had a mark of 75-68-21 in two seasons with the Islanders.