The Real Madrid playmaker scored a second-half double as Joachim Low's side put their European Championship misery behind them with a dominant performance against the minnows
Joachim Low's side dominated the match from start to finish, but a mixture of poor finishing and excellent goalkeeping from Gunnar Nielsen limited them to one goal, scored by Mario Gotze at the break.
The second-half was a slightly easier affair for the hosts, who extended their lead through a Mesut Ozil double and ran out comfortable winners.
Unsurprisingly, it was the hosts who made all the early running. Thomas Muller's charge into the box was unceremoniously ended by Rogvi Baldvinsson, but Ozil could only lift the ball into the wall.
Soon after, Nielsen began to shine. Miroslav Klose, chasing down Gerd Muller's international record of 68 goals, had an effort saved from point-blank range, before the Manchester City keeper acrobatically pawed away a Holger Badstuber cross that was creeping into the top corner.
Philipp Lahm sent a fantastic ball into the boc, which Klose met, but the Lazio veteran could only head the ball onto the roof of the net from a matter of yards, while Muller had a close-range effort blocked after a winding run down the right from Gotze.
Marco Reus thought he had done enough to break the deadlock on 22 minutes when he sent Lahm's pass towards the bottom corner, but once again Nielsen was there to turn the ball around the post.
Klose had already squandered another decent opportunity when the breakthrough finally arrived. Gotze picked up the ball about 25 yards from goal, beat a string of defenders before passing the ball into the bottom corner of the net.
However, falling behind did nothing to dampen the visitors, who refused to capitulate to their illustrious opponents. Klose wasted yet another opportunity to get on the scoresheet after some fantastic work from Marco Reus.
Baldvinsson prevented a certain goal when he threw himself in front of Sami Khedira's goal-bound shot before Muller came agonisingly close to notching his first international goal in almost a year with a header after Reus' acrobatic cross.
The Faroes had been resolute at the back, but going forward, they had offered next to nothing, with their only moment of note in the first-half coming on 13 minutes when the ball dropped to Simun Samuelsen just inside the Germany box, but the 27-year-old miscued his volley.
Germany made a resurgent start to the second-half and after the visitors won a rare corner, which was cleared with ease, Ozil picked up the loose ball, sprayed it out wide to Muller, and bombed into the area to volley home the Bayern Munich man's right-wing cross.
The hosts were cruising and looked capable of delivering a third at any moment. Nielsen did well to claw a cross away from Klose's head, and Khedira's ferocious rebound was only blocked by the heroic Hallur Hansson, who was left dazed.
The visitors' goalkeeper got down low to turn a 20-yard Ozil shot around the post, before the Real Madrid playmaker then laid the ball into Andre Schurrle's path, only for the Leverkusen man to take a heavy first touch, when he looked set to score.
The third finally arrived with just under 20 minutes to play. Reus laid the ball into Ozil's path, and the former Werder Bremen man kept his cool and slotted the ball home, despite Nielsen's best efforts.
The Faroese defence remained defiant til the last, making goal-saving blocks to deny Schurrle, Lukas Podolski and even Lahm, before Podolski's first touch let him down in a promising area.
Neuer remained aware enough to block Christan Holst's long-range effort and gather it at the second attempt and deny the visitors a consolation goal.
Germany coasted through the remaining minutes of the game and ended their two-game losing streak. They make the short trip to Austria on Tuesday.