Back-to-back defeats against Spain in their previous two outings had rendered this match a must-win affair and things started badly when they fell behind to a Vladimir Voskoboinikov striker after seven minutes.
Tuncay got his first on the half-hour mark before Sercan Yildirim added a second with 36 minutes played. Estonia were not to be denied and drew level through Konstantin Vassiljev just after half-time, but goals from Arda Turan and Tuncay's second secured three vital points.
Turkey remain third, four points behind second-placed Bosnia-Herzegovina, who they meet in a crunch match on Wednesday. Estonia's early goal in seventh minute shocked the home fans when Ragnar Klavan's low cross from the left allowed Voskoboinikov to tap in from close range.
Turkey struggled to find their rhythm despite Turan creating several opportunities. After 16 minutes Turan showed quick feet to get past two defenders on the left wing, but his low cross was intercepted by Estonian goalkeeper Sergei Pareiko with Yildirim lurking.
Turan then lashed a shot from outside the area wide of the right post. Tuncay equalised after 29 minutes following a one-two with Emre Belozoglu down the left wing, which freed the Stoke striker to fire home from six yards.
The hosts went ahead soon after when Turan fought his way into the area and while his shot hit the post it rebounded to Yildirim whose follow-up shot deflected in off Raio Piiroja. Turkey conceded their advantage just seven minutes after the restart when Vassiljev's right-footed long-range shot found the net.
It was Turan who deservedly restored his side's advantage after 62 minutes when he headed Hamit Altintop's corner to the far post home. Hamit then set-up Tuncay's second with 18 minutes remaining, slipping through a pass that the striker lashed into the bottom-right corner.