A defensive lapse by the hosts early in the second half allowed Germany's Thomas Mueller to find Klose unmarked just inside the Belgian penalty area and the German marksman made no mistake.
Germany, perhaps still shaking off post-World Cup lassitude, were slow to shine and came off second best for large patches of the first half. The young Belgian side refused to sit back and twice forced German keeper Manuel Neuer to save shots from 17-year-old striker Romelu Lukaku and Moussa Dembele.
However, the World Cup semi-finalists dominated after the break and effectively closed down the game after Klose's goal.
For many of the German players it was their first match for their country since their 1-0 loss in South Africa to Spain. Germany put out a largely second-string side for their 2-2 friendly draw with Denmark last month.
Belgium have now lost eight straight matches against Germany and West Germany since a 0-0 draw in 1982, with their last victory in 1954.
Germany host Azerbaijan on Tuesday. Belgium, who have not qualified for a major championship since the 2002 World Cup, travel to Turkey, who lead group A after a 3-0 opening win over Kazakhstan.
Belgium's coach Georges Leekens and his players smarted at the loss, many saying a draw might have been a fairer result.
"The Germans settled it with a gift from us," Leekens said. "They were more professional. We still have something to learn."
However, he said he was encouraged by the passion of his team, with nine of the starters 24 years or younger.
German coach Joachim Loew was also complimentary about the hosts: "It is important to win away in Belgium. I think other teams will drop points here," he said.