Lagerback, who managed Sweden for nine years, takes over from Per-Mathias Hogmo, who parted ways with Norway after three lacklustre years in charge following a disappointing start to the qualifying campaign for the 2018 World Cup in Russia.
"It feels very exciting," said 68-year-old Lagerback, who had taken a role as advisor to new Sweden boss Janne Andersson.
Norway failed to qualify for the 2014 World Cup and the Euro 2016 tournaments and their FIFA ranking has plunged to an all-time low of 84th in the world, down from 24th in 2012.
They are second-bottom of World Cup qualifying Group C with three points from four games behind Germany, Northern Ireland, Czech Republic and Azerbaijan after one win over San Marino.
The pragmatic Lagerback was in charge of Nigeria at the 2010 World Cup as well as his long spell with Sweden but perhaps his greatest achievement was helping Iceland, with a population of just over 300,000, to reach their first major tournament.
At the European Championship finals, Iceland qualified from their group and produced a huge shock by beating England in the last 16 before going out in the quarter-finals to hosts France.
(Reporting by Philip O'Connor in Stockholm; Editing by Ken Ferris)
