There was early drama as Faroes goalkeeper Gunnar Nielsen kept the hosts at bay for almost half an hour with some heroic saves and a hint of desperation was creeping into the Germans' play before Mario Gotze broke through.
After that Germany relaxed, stroked the ball around, often at walking pace, and completed the win with two Ozil goals in the second half.
The game was largely played in near silence, broken only by a monotonous drum-banging in the background, and even the goals brought only muted cheers from the crowd.
Germany coach Joachim Low said his team were still wasting too many chances, one of their weaknesses as Euro 2012 in which they lost in the semi-finals.
"We had five or six really good chances in the first half and our problem was that we missed too many of them, something we have done in other games," said Low.
Ozil said patience had been important.
"We wanted an early goal and we missed chances we would normally have put away but we must be satisfied with the result," he said.
Nielsen made excellent saves to defy Sami Khedira and Marco Reus as the opening exchanges resembled an attack-versus-defence training session.
Gotze settled German nerves in the 28th minute when he collected the ball around 30 metres from goal, weaved his way between the Faroe defenders and rolled the shot just inside the post from 12 metres.
A rare Faroes Islands attack led to Germany's second in the 54th minute. The hosts broke out of defence, Thomas Muller crossed from the byline and Ozil side-footed the ball into the net from 12 metres.
The Real Madrid midfielder provided another neat finish in the 72nd minute when Reus slipped the ball to him and he clipped the ball past Nielsen from inside the area.
