The forward capitalised on a defensive mix-up and fired into an empty net in the 35th minute to put his team ahead after a clever pass by Mesut Ozil left Russia goalkeeper Igor Akenfeyev off his line. It was Klose's 50th goal for Germany.
Germany, who have 25 points from nine games, will finish top of the group with second-placed Russia, on 21 points, still able to join them in next year's finals with a win in the playoffs.
"I was sure all week the team would be focused and would play to win," said Germany coach Joachim Loew. "No team in the world can play for a draw, and that certainly includes us.
"We said before the match we had one hand on the ticket to South Africa and we'd be able to book our ticket ourselves. I think tonight they'll be allowed to celebrate a bit."
"Adler showed a lot of calm back there and incredible reflexes," said Loew. "He made a very good impression all in all. I'd pay him a big compliment for this performance."
"Of course I'm disappointed with the result tonight but this team showed great character, especially in the second half," Hiddink told a news conference.
"We missed several good chances in the second half and I think a foul on Arshavin was 100 percent a penalty.
"We also weren't good enough with the final pass but if we play like that we have a good chance to qualify in the playoff in November," the Dutchman added.
