The victory in Torshavn means that Joachim Low's side are now five points clear of second-placed Sweden at the top of Group C with two more rounds of matches remaining.
Then in the 10th minute Miroslav Klose tested Gunnar Nielsen with a powerful shot before Julian Draxler saw his attempt saved by the Faroes goalkeeper.
Germany eventually surged ahead midway through the first half when Per Mertesacker turned home from a corner just moments after Klose had been denied by the post.
The Lazio striker came close again on the half-hour mark but was kept out by Nielsen.
The Germans continued to create chances in the second half, with Ozil, Draxler and Khedira all coming close, the latter being denied only by the post on 68 minutes.
Then in the 74th minute Ozil doubled Germany's advantage from the penalty spot. Atli Gregersen brought down Thomas Muller inside the box and was given his marching orders, and the Arsenal playmaker slotted home from 12 yards.
Muller himself wrapped up the scoring six minutes from time, lashing home Philipp Lahm's cross from close range.