An early Ragnar Klavan own-goal set the hosts on their way to a one-sided win, with Tamas Hajnal, Daniel Bode, Krisztian Nemeth and Balazs Dzsudzsak also on target. Tarmo Kink replied for Estonia early in the second half.
The win moves Hungary into pole position to claim second place and a play-off berth behind Group D winners the Netherlands, who secured their passage to Brazil with victory in Andorra tonight.
Romania's 2-0 defeat at home to Turkey allowed Hungary to move above them by a point with two matches remaining. Hungary travel to the Netherlands for their next preliminary game in October, before finishing their campaign at home to Andorra.
The hosts came into the match still hurting from Saturday's 3-0 defeat in Romania and took the game to Estonia from the outset, Nemeth striking a shot against the bar inside five minutes.
The lead was not much longer in coming, with just 11 minutes on the clock when Klavan turned the ball into his own net.
Hungary's advantage was doubled on 21 minutes when Hajnal volleyed sweetly into the right corner after connecting with Vladimir Koman's cross from the right.
And it was 3-0 after 41 minutes as Bode took aim from distance to fire powerfully beyond Sergei Pareiko in the Estonia goal.
The visitors rallied after the restart with Kink curling home a cute finish inside three minutes, but the three-goal cushion was restored with 20 minutes to go as Koman played in Nemeth for a powerful finish from the edge of the area.
The win was wrapped up with six minutes left when Dzsudzsak curled home a free-kick.