The SFA moved for the former Celtic boss after sacking Craig Levein in November following a disastrous run of results.
The 55-year-old, an inductee of the Scotland Hall of Fame after amassing 50 caps for his country, has been out of management since leaving Middlesbrough in October 2010.
Strachan's first match will be a friendly against Estonia at Pittodrie early next month before a World Cup double header against Wales at Hampden and a trip to Serbia on 22 and 26 March.
Strachan, who has a deal until the end of the European Championships of 2016, said: "I am very, very proud. Proud for myself and proud for my family and friends.
"It shows you how important the Scotland job is and it is a great day for me to be able to take up this opportunity. It's been 40 years in the making but it has been worth it.
"I know I face challenges, on behalf of the nation, but I hope to be able to make this nation proud... with help from other people.
"I believe the time is right for me to make the move into international football as I have had the European experience and how to deal with players.
"I always wanted to be Scotland manager and, with the help of a lot of other managers and players, I am here today."
SFA chief executive Stewart Regan said: "Gordon made no secret of his desire for this job and we are delighted he is now in charge of the national team."
During his playing days, Strachan was a member of the successful Aberdeen team of the late 70s and early 80s before moving to Manchester United in 1984.
After a five-year stay at Old Trafford, he was transferred to Leeds where he won the league championship in 1991-92.
In his managerial career, he had spells with Coventry, Southampton and Celtic - where he won three successive SPL titles - before moving to Middlesbrough in 2009.
Scotland have not qualified for the finals of a major tournament since reaching the 1998 World Cup in France.
"To progress with the squad. I really want the squad and the staff to give something back to the country and the fans who support us because the fans are probably more famous than the squad now," Strachan added.
"So what we want to try and do is to give back something and make them turn up for a major finals competition."
Scotland have picked up just two points in their bid to qualify for the World Cup in Brazil, but Strachan rejected the suggestion that a top-two spot is out of his players' reach.
"If we work together as a group and a set of fans I know for a fact we'll be successful," he added. "I can't ask for anything more than we're doing at the moment.
"We're going to use the games to try and win. My philosophy is to win games of football, like Manchester United.
"We still have to collect as many points as we can and it's disrespectful to say we'll use the games just to improve. We'll give it a go.
"What I know is that these guys, for all they might get criticised, are the best at what they do. They play for Scotland and we have to find a system that suits these players to win games of football."