Scotland get off to winning start under Strachan
Charlie Mulgrew scored the only goal of the game in the first half, steering home a Charlie Adam free-kick in a well-worked training ground routine.
It was Strachan's first game in charge since replacing Craig Levein in January and the former Celtic boss will have been pleased with the effort his players showed in grinding out a result against a dogged Estonia.
Strachan had never won his first match in charge in any of his previous managerial jobs, but his team never looked in any danger of slipping up after they took the lead on 39 minutes.
Stoke City midfielder Adam shaped to cross a freekick to the far post but instead squared it low to Mulgrew who steered it home left-footed.
Scotland are currently bottom of World Cup qualifying Group A with two points from four games.