Celtic's Charlie Mulgrew struck in the first half, sweeping a Charlie Adam free-kick home from close range.
Scotland were good value for the win, with substitute Jordan Rhodes going close in the second half.
The new Scotland manager made eight changes to the side that caretaker Billy Stark had named for the win over Luxembourg in November.
Estonia allowed Scotland to dictate the opening stages, giving the hosts confidence despite the tricky playing surface. Shaun Maloney's close-range volley forced Sergei Pareiko into an early save after the Estonia goalkeeper had failed to deal with a corner, serving notice that set-pieces could be productive.
Chris Burke, back in the national side for the first time since 2006, tormented Taijo Teniste on the right flank, but it was the Estonian who served notice that the visitors were not at Pittodrie just to make up the numbers, forcing Allan McGregor to push a shot from distance over the bar.
Having weathered Scottish pressure, Estonia gradually applied themselves in attack with a teasing cross from Henrik Ojamaa of Motherwell deserving better than to skim across the face of the goal.
Teniste lashed another shot over the bar, but at the other end Steven Fletcher had nodded a looping Burke cross over before Maloney forced a save.
Estonian attacks were rare, but McGregor was alert enough to dive at Tarmo Kink's feet after a lifted ball from Konstantin Vassiljev had foxed the Scottish defence.
With a tough pitch and determined opponents to deal with, set-pieces looked like they would be the main goal threat for Scotland, and so it proved with six minutes to the break.
Adam whipped in a low free-kick from the left, which Mulgrew drove home first-time from around 12 yards out, having shaken off the Estonian defence. The Celtic left-back, who left Parkhead while Strachan was in charge of the Bhoys, was delighted with his first international goal.
Strachan made a raft of substitutions in the second half to assess some of his other options with Wales and Serbia lurking in the World Cup qualifiers next month. Of his replacements, Rhodes in particular looked sharp, firing over from the edge of the box and in the 64th minute a similar effort from Steven Naismith also sailed over.
Mulgrew looked keen to double his tally and in the 71st minute his powerful drive from 25 yards was tipped over by Pareiko.
The game stuttered in a disjointed way into its final stages, with Estonia enjoying their best spell in the final quarter. Substitute Gert Kams fired wide of the target late on to cause some concern in the face of Scotland's still narrow lead.
But the Scots held on to give hope ahead of March's tougher tests in the unlikely pursuit of qualification for Brazil in 2014.