Germany have maintained the momentum from their third place World Cup finish last year, winning all five of their qualifiers to top the group with 15 points, conceding only one goal and scoring 17 in the process.
Klose scored in his sixth consecutive international, tapping in from close range after three minutes and 2010 World Cup top scorer Muller struck twice before the break, both times set up by Real Madrid's Mesut Ozil.
"It was hugely important to carry that moment of the World Cup into this competition and we have done that," said Muller. "It was not always easy or beautiful to see for the fans with their defending but we did our job well."
"The first half was good and we had a handful of chances but in the second half we took our foot off the gas," said coach Joachim Low.
With nine players from Germany's World Cup team in the starting line-up and with the visitors missing six key players it was always going to be a one-sided affair.
Klose added his second goal two minutes from time with a tap-in to move seven goals short of Gerd Muller's Germany record of 68 international goals, although the former Bayern Munich striker scored them in just 62 games.
