O'Driscoll, who had not taken charge of a single competitive game at the Red Devils, left the club late on Thursday evening.
“The club is extremely disappointed to be losing Sean so soon after he came to Crawley Town," said chief executive Alan Williams.
“But our policy has never been to stand in the way of any employee who feels they would prefer to pursue a role at a different club and following lengthy and intensive negotiations we have been able to agree compensation with Nottingham Forest."
O'Driscoll spent five months as first-team coach as the City Ground last season. He is the second manager to leave the club in 2012, after Steve Evans left to become Rotherham boss.
Steve Coppell, the club's director of football, will take charge of the first-team in a caretaker capacity along with head coach Craig Brewster.
Forest sacked Steve Cotterill last week following the completion of a takeover by the Kuwaiti businessman Fawaz al Hasawi, saying they wanted an iconic person to replace him.
Mick McCarthy, the former Wolves manager, was linked with the role but al Hasawi is confident he got the right man.
"Sean's passion for Nottingham Forest, knowledge of the game and the Championship in particular, plus his work ethic, shone through in our conversations," he said.
"He is hugely respected, not only by members of our current squad and staff at the Club for his work in helping transform our fortunes last season, but also within the wider football community."
O'Driscoll has previously managed Bournemouth and Doncaster, he was appointed Crawley boss in May.