The Icelandic Football Association confirmed the appointment on their website on Friday, with the 63-year-old due to officially take over on January 1 next year.
Lagerbäck coached his native Sweden for more than a decade and also led Nigeria at last year’s World Cup in South Africa.
He will now lead Iceland in the qualifying campaign for the 2014 World Cup in Brazil.
“I like Iceland a lot… I like the country and I like the people I’ve got to know here,” Lagerbäck told AFP following the announcement of his appointment.
“I have a lot of respect for what (Iceland is) doing as a small country in producing footballers and what (it has) done through the years.”
Lagerbäck acknowledged that he is not very familiar with the current Icelandic team, but insisted that time was on his side.
“I start in January and will get to know all the players and hopefully play three or four friendlies before we play any qualifying games,” he said, pointing out that “that’s better than most national team coaches get.”
The island nation has never before qualified for a major tournament, but Lagerbäck believes they can negotiate a World Cup qualifying group that contains Norway as top seeds, and does not feature a single country that will feature at Euro 2012.
“I really want to win. That’s the most important thing for me,” he said.
“I mean we are not in the toughest group for the World Cup. We don’t have any of the five or six biggest countries in Iceland’s group, but at the same time it’s always difficult to qualify.
“I will do my absolute best and hopefully get a lot of response from the players so we can do it together and get really good results. That’s what I’m hoping for,” he said.
Neither the Icelandic Football Association, nor Lagerbäck himself, were willing to comment on what his salary would be.
“I don’t want to talk about it,” he said, adding only that “I can assure all the Icelandic fans and footballers that I don’t think I’m ruining the Icelandic FA’s economy.”
“It’s a good salary of course, but if I wanted to get money, I would have stayed in Sweden.”