Mertesacker gave Germany the lead midway through the first half, but they then had to wait until midway through the second half before a penalty from Mesut Ozil doubled their 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.
The win came after a workmanlike performance, which caused Mertesacker to demand a more decisive performance.
"We knew it would not be easy in the Faroe Islands and on the artificial turf," Mertesacker told kicker.de.
"In the decisive moments, how we played in attack was not so consistent.
"We must do better."
Coach Low echoed the Arsenal defender's sentiments.
He said: "It was a mandatory task to pick up three points here.
"It is the final third which we still have to work at. We need to improve our game in the final third still."
National team manager Oliver Bierhoff added: "We could have scored a goal or more after the break, we lost a little bit of rhythm, but in the end the victory was important."