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 club-mate Per Mertesacker has 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, 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.
Miroslav Klose was next to threaten the Faroe Islands goal, but his low drive in the 10th minute was saved by the legs of Nielsen.
Midway through the half, Klose was inches away from scoring a record-breaking 69th international goal for Germany.
After patient build-up 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 were not to be denied for long as, from the resulting corner, Jerome Boateng flicked the ball on at the near post and Mertesacker prodded home from two 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 goalscorer, as his header from Marcel Schmelzer's cross was gathered low to his left by Nielsen in the 30th minute.
Nielsen was called into action once again three minutes later, beating away Draxler's fierce effort following a short corner from Toni Kroos.
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.
Draxler created an opening for himself on the hour mark with some quick feet inside the area, but his left-footed effort lacked power and was easy for Nielsen to deal with.
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 their dominance merited.
With 10 minutes to go, substitute Andre Schurrle, on in place of Draxler, fired just over the angle of bar and post from outside the area as Germany looked to give the scoreline a more convincing gloss.
Muller did just that four minutes later, firing home from Lahm's cut-back from the right as the hosts visibly tired.Germany currently lead Group C by five points with two games remaining and look a safe bet to progress to the finals in Brazil in 2014, with Sweden's earlier win over Kazakhstan denying them qualification on Tuesday.