The Arsenal star struck a penalty in between goals from Per Mertesacker and Thomas Muller to put the Nationalelf on the brink of reaching next summer's finals.
The new Arsenal signing struck midway through the second half after teammate Per Mertesacker had put Joachim Low's side in front, with Thomas Muller sealing the three points in the closing minutes.
The hosts made three changes from the team that lost 2-1 in Kazakhstan on Friday, with Atli Gregersen, Pol Johannus Justinussen and Daniel Udsen replacing Jonas Naes, Hallur Hansson and Christian Mouritsen. Germany had just one change from the side that defeated Austria 3-0 on Friday, with Julian Draxler coming in for Marco Reus, who was absent with a stomach infection.
The visitors fired the first warning shot in the fourth minute as Sami Khedira sent a low drive narrowly wide of Gunnar Nielsen's right-hand post, before Suni Olsen deflected the ball onto his own crossbar from Draxler's cutback.
The hosts responded immediately, with Justinussen breaking down the left and letting fly with an effort from the corner of the box that Manuel Neuer had to be alert to tip over.
Midway through the half, Miroslav Klose was inches away from scoring a record-breaking 69th international goal for Germany.
After patient buildup play involving Muller and Philipp Lahm, Klose struck the base of the post from 10 yards before Gregersen hacked the rebound over his own crossbar.
But Germany was not to be denied for long. On the resulting corner, Jerome Boateng flicked the ball on at the near post and Mertesacker prodded home from 2 yards out.
Klose continued to search in vain for the goal that would take him clear of Gerd Muller as his country's all-time leading goal-scorer, as his header from Marcel Schmelzer's cross was gathered low to his left by Nielsen in the 30th minute.
In the second half, Germany continued to press for a second goal, with Ozil curling a free kick narrowly wide from 30 yards in the 48th minute. Khedira almost grabbed a second in the 67th minute, rising at the far post to meet Draxler's cross from the left, but his downward header was cleared off the line by Rogvi Baldvinsson.
Five minutes later, Nielsen made hard work of denying the visitors once again, fumbling an Ozil shot before diving at Draxler's feet to smother the rebound.
A difficult night for the hosts was made even harder in the 73rd minute as Gregersen was dismissed for bringing down Muller inside the area as he bore down on Nielsen, and their resistance was finally broken.
Ozil stepped up and calmly sent the goalkeeper the wrong way as Germany at last had the second goal its dominance merited.
With 10 minutes to go, substitute Andre Schurrle, on in place of Draxler, fired just over from outside the area as Germany looked to give the score line a more convincing gloss.
Muller did just that four minutes later, firing home from Lahm's cutback from the right with the hosts visibly tired.Germany currently leads Group C by five points with two games remaining and looks like a safe bet to progress to the finals in Brazil in 2014, with Sweden's earlier win over Kazakhstan denying it qualification Tuesday.