Per Mertesacker gave Joachim Low's men the lead midway through the first half, but Germany had to wait until midway through the second half before a penalty from Arsenal's Mesut Ozil doubled the advantage from after Atli Gregersen was sent off.
Thomas Muller, who had earned the penalty, then rounded off the win in the 84th minute as Germany maintained a five-point lead over Sweden at the top of Group C.
It can book its ticket for the FIFA World Cup with a win over Republic of Ireland next month while a point would be enough if Sweden, its final group opponent, fails to beat Austria in its next game.
The crossbar came to the home side's rescue in the fifth minute when Suni Olsen almost turned Julian Draxler's cross into his own goal, but there was then a scare for Germany goalkeeper Manuel Neuer at the other end when he tipped Pol Johannes Justinussen's shot over.
It was an indication that Faroe Islands was not just going to build the barricades and accept an eighth straight group-stage defeat without a fight.
However, the only real chances of note came at the other end of the field with Miroslav Klose going close in the 10th minute, when his angled shot was saved, and then even closer in the 22nd minute when he hit the inside of the post.
Less than a minute later, Germany took the lead with its much-maligned defence combining to score from a corner.
Jerome Boateng met the ball at the near post and nodded it on, just beyond the reach of Klose, but on a plate for central defensive partner Mertesacker, who touched the ball over the line from a yard out.
Low's side struggled to add to the lead with Sami Khedira coming close midway through the second half, seeing his header cleared off the line, before a penalty helped the visitor on to the points.
Muller was put through on goal and he went down on his way past Gregersen, who was shown the red card for a foul inside the penalty area.
Ozil stepped up to send Gunnar Nielsen the wrong way and establish a more comfortable lead for Germany.
Victory was sealed six minutes from time when Muller picked up Philipp Lahm's cross in a rather ungainly fashion, but still managed to get in a shot which deflected past Nielsen.