In a fairly unremarkable encounter at Pittodrie in Aberdeen, the poor condition of the pitch took centre stage, while a raft of substitutes slowed the play in the second half
After chances for both sides in the early stages, Charlie Mulgrew gave Scotland the lead seven minutes before half-time, but little happened from then until the final 20 minutes.
Mulgrew went close again late on and Estonia had a few opportunities, but no players were able to find the net and the score remained 1-0.
Strachan named a fairly strong starting line-up as the likes of new Mallorca signing Alan Hutton, Charlie Adam and Steven Fletcher all featured.
The visitors fielded a team full of relative unknowns, but record goalscorer Andres Oper and Henrik Ojamaa of Motherwell were both present.
Throughout the match Pittodrie's pitch was a bit of an issue, preventing the teams from tying together extended passing moves, but the hosts did manage to carve open an early chance.
Wigan Athletic man Shaun Maloney did well on the edge of the area before firing a shot goalwards, but Sergei Pareiko got down well to make the save.
Adam went close from inside the area four minutes later, while Alan McGregor was soon forced into a good save as defender Taijo Teniste let fly from 30-yards, but he could not beat the goalkeeper.
After several chances in the opening 10 minutes or so, the match soon settled and a clash of heads saw Christophe Berra struggling for a short period.
Around the half-hour mark Teniste tried his luck from range again, but this time it sailed well over the bar and Scotland soon had the lead.
Estonia had made a habit of giving away pointless free kicks around their area and eventually Scotland made them pay as Mulgrew converted a chance following a well-worked set-piece from Adam to give them the lead at half-time.
The tempo of the match was visibly affected after the break as the hosts made a number of substitutions, but Robert Snodgrass should have controlled Steven Naismith's pass better when played into a dangerous position.
It was not long before Estonia made some changes of their own, also to the detriment of the match, but Naismith did have a good chance with 30 minutes to go.
The Everton mad dropped his shoulder and beat his man on the edge of the box before unleashing a fearsome drive at goal, but it fizzed just over the bar.
Mulgrew soon went close to getting his second goal of the match as he also took aim from the edge of the Estonian penalty area, but Pareko made another good stop.
In the final few minutes, Berra failed to deal with a lofted cross allowing Sander Puri with the chance to volley goalwards, but the midfielder missed the target and Scotland held on for the win.