Solskjaer spent 11 seasons at Old Trafford, scoring the winning goal in the 1999 Champions League final.
“Manchester United is in my heart and it’s brilliant to be coming back in this role,” said Solskjaer.
“I’m really looking forward to working with the very talented squad we have, the staff and everyone at the club.”
Solskjaer, who earlier this month signed a new deal as manager of Norwegian club Molde, will be joined by Mike Phelan, who returns as an assistant manager having previously working alongside Sir Alex Ferguson.
Michael Carrick and Kieran McKenna, both part of Mourinho’s coaching staff, will continue to work under Solskjaer.
Solskjaer’s first match in charge will be against Cardiff – where he endured an ill-fated spell as manager – on Saturday.
The 45-year-old takes over with United sixth in the Premier League.