Celtic defender Mulgrew got the only goal in the 38th minute at the home of his former club when he fired in from a Charlie Adam free-kick - his first goal in dark blue.
Ironically, when he was boss at Celtic, Strachan deemed Mulgrew surplus to requirements during the defender's first stint at Parkhead.
It was a solid if unspectacular start to Strachan's reign in a bitterly cold night in Aberdeen with Birmingham winger Chris Burke, back in the fold for the first time in seven years, combining well down the right with his former Rangers team mate Alan Hutton until he was left inside at the interval.
Qualification for the 2014 World Cup in Brazil maybe a forlorn hope - the Scots have two points from four games in their qualifiers - but this will give them optimism ahead of qualifiers against Wales and Serbia next month.
Strachan picked a positive-looking side on the return to the ground where he had been a favourite as a player.
Steven Fletcher led the line backed up by Burke, Shaun Maloney and Steven Naismith with skipper Scott Brown and Adam providing the ballast in midfield.
A back four of Alan Hutton, Christophe Berra, Andy Webster and Mulgrew, in front of goalkeeper Allan McGregor, completed the line-up while the visitors had one familiar face in Motherwell's Henrik Ojamaa.
Scotland, who had not lost to Estonia in seven previous encounters, started confidently and in the third minute visiting keeper Sergei Pareiko made a decent save from Maloney at his front post.
Burke was then released down the right channel by Hutton but he turned back into traffic and his shot was blocked.
However, with Estonia's first attempt on goal in the ninth minute, Taijo Teniste's arrowed drive from 30 yards was heading for the top corner until McGregor stretched to tip the ball over the bar.
Hutton and Burke combined well down the right again in the 20th minute and when the latter stood a ball up at the back post Fletcher headed just over the bar.
Moments later, Maloney picked up the ball and slalomed past Sergei Mosnikov and Igor Morozov but his shot from the edge of the box was parried by Pareiko with Enar Jaager clearing to safety.
Estonia countered with another decent long-distance drive from Teniste which cleared the bar by a yard and in the 31st minute a lovely lofted pass by midfielder Konstantin Vassiljev beat the Scotland defence and McGregor had to charge from his line to block from Tarmo Kink.
Scotland's opener looked as if it had been crafted on the training ground and came when the quick-footed Maloney was fouled outside the box, on the left-hand side, by Morozov.
Adam shaped to curl the free-kick high into the middle but instead knocked it low along the ground to find Mulgrew, who had peeled away from the crowd - and the Hoops player fired a left-footed shot past Pareiko from 12 yards.
The goal lifted the crowd and the players and moments later Hutton's cross from wide on the right almost caught out Pareiko at his near post with the Wisla Krakow keeper having to concede a corner, with Berra heading Mulgrew's delivery wide.
Jordan Rhodes and Robert Snodgrass replaced Maloney and Burke respectively for the start of the second-half and a whole raft of changes for both sides followed, with James McArthur, James Morrison, Kenny Miller and Kris Commons also getting into the action.
Rhodes looked sharp, firing over from the edge of the box and in the 64th minute a similar effort from 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 for a corner, which came to nothing.
The game stuttered in a disjointed way into its final stages, although there was a moment of brief panic among the home fans before Estonia substitute Gert Kams fired wide of the target.
However, the Scots were mostly in control until the final whistle signalled a successful start to Strachan's tenure - although there will be far tougher tests to come, and soon.