With two games remaining in Group C, Germany has 22 points, five ahead of Sweden which won 1-0 in Kazakhstan thanks to Zlatan Ibrahimovic's first-minute strike.
Ibrahimovic beat Kazakh goalkeeper Andrei Sidelnikov with a crisp low shot inside the far post.
Austria defeated Ireland 1-0 in Vienna to stay three points behind Sweden and all but rule out Irish participation in Brazil next year.
"Sweden won so it's not possible for us anymore to think about the possibility of qualifying," Ireland coach Giovanni Trapattoni said.
David Alaba scored in the 83rd minute when he hammered home from short distance after several Ireland defenders failed to clear the ball.
"Ireland defended well but thank God we won the game," said Alaba, who scored the equalizer in injury time in the sides' first meeting, a 2-2 draw, last March. "We are all dreaming of Brazil and will do everything it takes to make this dream come true."
The game was in the balance until Marko Arnautovic came on for the final half hour. The Austria forward saw his header cleared off the line before Ireland 'keeper David Forde denied the unmarked Andreas Weimann.
Defenders John O'Shea and Richard Dunne picked up yellow cards and will miss Ireland's next game in Germany, where the hosts will aim to wrap up qualification.
The Faeroe Islands attacked from the start at the Torsvolli Stadium, but Germany quickly assumed control and Sami Khedira fired narrowly wide in the third minute.
Suni Olsen almost conceded an own goal only for the ball to rebound off the crossbar, before Germany 'keeper Manuel Neuer blocked Pol Johannus Justinussen's effort at the other end.
Miroslav Klose's first shot was saved by Gunnar Nielsen in the 10th, and the Germany striker's quest for a record 69th international goal was thwarted by the post in the 22nd.
Mertesacker scored his third when he came in at the far post to meet Jerome Boateng's header from Ozil's corner with his left boot.
The second half failed to yield as many scoring chances. Khedira came close again in the 67th, when Rogvi Baldvinsson cleared his header off the line, before Mueller went down and Ozil put the result beyond doubt.
"We were under obligation to win the three points. They made it difficult until they were down to 10 men," Germany coach Joachim Loew said. "We had six or seven good opportunities in the first half but there wasn't as much movement in the second."
Loew, whose side had been criticized for its shaky defense in recent friendlies, said the offense was not as good as it should have been.
"We have to improve our game in the final third," Loew said. "We generally win one or two nil against the smaller teams, or by a lot of goals we always win but we have to make more of the chances we're creating."
