The Scots last got through to a major tournament when reaching the 1998 World Cup, and Strachan took time to consider his future after his side finished fourth in their European Championship qualifying group, behind Germany, Poland and the Republic of Ireland.
But the vocal backing of an 11,000-strong travelling support in Scotland’s final qualifier, a 6-0 win against Gibraltar, helped persuade Strachan to stay on.
Strachan will now begin work on preparing his squad for a World Cup qualifying group involving England, Slovakia, Slovenia, Lithuania and Malta.
“I want the best for our country and for those players and I believe we can make Scotland proud in the upcoming qualifiers,” the 58-year-old said.
“For that reason, I am delighted to have been asked to continue in my role and I look forward to working with my backroom staff and the players to improve on the previous campaign and look forward to the World Cup qualifiers with optimism.”
Strachan will again be assisted by Mark McGhee, who took over as Motherwell manager on Tuesday, and former Fir Park boss Stuart McCall.
Strachan said: “It is an honour and a privilege to manage my country and, in a way, that has made the disappointment of not qualifying for Euro 2016 all the more profound.
“None the less, the reaction from the supporters throughout the campaign, and in particular from those who travelled to Faro in their thousands, is an experience that will live with me for the rest of my life. It was a recognition that the players’ efforts had been appreciated.
“That appreciation and goodwill has also been apparent in the messages of support I have received this week, from emails and letters to chats in the street.”
SFA chief executive Stewart Regan added: “The board of the Scottish FA has been consistent in our belief that Gordon should continue the good work that has been apparent in many performances and results throughout the previous campaign.
“Similarly, the supporters and the players have been overwhelming in their view that Gordon is the national coach to lead us into the World Cup qualifying campaign.
“The mutual respect and stability that exists is essential to our future success and we all look forward to planning for Russia 2018.”