The ex-Norway international scored 126 goals for United before retiring in 2007 due to a knee injury. He then moved on to the coaching staff at Old Trafford and has been in charge of the reserve team.
"I always said I wanted to go home and properly start my career as a manager," said the former "baby-face assassin" who remains youthful-looking at 37.
Solskjaer added that he took the job at Molde, where he played for a year and a half in the mid-1990s, with the aim of lifting them from mid-table and eventually winning silverware.
"From United I've brought knowledge about how a club should be run including the importance of the youth division," he told a news conference on Tuesday.
United manager Sir Alex Ferguson said it was the right time for Solskjaer to take the plunge and become a first-team coach.
"Ole leaves with our best wishes. He's been a fantastic servant and he's contributed to the development of our young players," said Ferguson.
"I think his career as an outstanding professional footballer gave him a foundation to get into coaching. He's had a great introduction with our reserves and this is the right time for Ole to take up his first managerial position.
"Ole will do well at Molde. It's a good start for him," added Ferguson.
Molde finished 11th in the 16-team top-flight last season, 28 points adrift of champions Rosenborg Trondheim.
