Crosby, who guided Sunderland to the FA Cup final in 1992, joined the club in 2004 as reserve team coach under Steve McClaren, and was promoted following Southgate's elevation when McClaren left to take up the England job.
Crosby: "Gareth Southgate has decided to make changes at the club and, unfortunately, I am part of those changes.
"I am very disappointed after working for the past three years as Gareth's assistant.
"But I would like to take this opportunity to thank everyone at the club, from the backroom staff, players and coaches, for making my time at Middlesbrough an enjoyable one.
"I would like to wish them all the best for next season."