Tamas Hajnal netted the game's only goal in the first minute of the second half as Hungary, beaten 4-1 at home by the Netherlands in their last match, secured their second victory of the qualifying campaign.
The result kept the Hungarians within three points of pool leaders the Dutch and Romania, and left Estonia still searching for their first point and their first goal.
Stuttgart midfielder Hajnal went closest to making the breakthrough in the first half with a 26th-minute free-kick which looked destined for the bottom corner before Sergei Pareiko tipped it around the post.
He did find the breakthrough seconds after the restart, though, lashing a right-footed shot into the top-left corner from Balazs Dzsudzsak's cross.
The Estonians, well beaten by Romania and Turkey in their opening two qualifiers, failed to mount a response, with Hungary goalkeeper Adam Bogdan never seriously tested throughout.