Germany, who have a point more that second-placed Russia ahead of their match in Moscow on October 10, were lacklustre for much of the first half, a shadow of the impressive team that beat South Africa 2-0 in a friendly on Saturday.
They found their form after Azerbaijan were left with 10 men from the 50th minute following a red card for Samir Abasov.
Striker Klose, a second-half substitute for Mario Gomez, fired in his first goal from close range on 55 minutes after captain Michael Ballack had converted a 14th-minute penalty.
The striker cleverly flicked on an Andreas Beck low cross 10 minutes later to make it 3-0.
"We made a very good start for about 20 minutes but after that I did not like how we played," said Germany coach Joachim Loew. "We only ran after them after the 1-0. I told my players so at halftime."
"Until now our qualification has been almost optimal, getting 22 out of 24 possible points," he said. "We will go to Russia well prepared."
Germany stormed forward, coming close on several occasions in the first 10 minutes. Striker Lukas Podolski's powerful left-footed shot bounced off Azeri keeper Kamran Agayev after five minutes and seconds later, midfielder Mesut Ozil shook off two defenders only to see his shot blocked.
Agayev, under siege for most of the first 20 minutes, made the save of the match when he punched striker Gomez's point-blank header over the bar.
But after grabbing the lead through Ballack, Germany stepped off the gas, allowing the Azeris, coached by former Germany coach Berti Vogts, to venture forward.
Klose's entry after the break turned things around, with the striker adding pace to their game. Podolski completed the rout in the 71st with Ballack heading the ball into the striker's path.
"We lost our rhythm after the first goal for some reason. We only ran after them," said Ballack. "But we got what we wanted and now comes the most difficult match of the whole qualification. We have to prepare well for the match against Russia."
