Germany dominated possession throughout but took several chances before substitute Heiko Westermann scored in the 28th minute, lifting a rebound past goalkeeper Kamran Agayev.
Playing in front of his Cologne home crowd, winger Podolski then scored in stoppage time and one minute later laid on an assist to Klose who took Germany to a 3-0 half-time lead.
Klose's second goal came in second-half stoppage time, taking his tally to 55 goals in 103 internationals, joint second with Joachim Streich who scored as many in 102 internationals for East Germany. Gerd Muller is Germany's all-time leading scorer with 68.
"I was happy to play in Cologne, in my living room," said Podolski who was substituted in Germany's 1-0 defeat of Belgium on Friday. "I have always said if... I'm fit, it's tough to get past me."
In the second half, Azerbaijan captain Rashad Sadhikov scored an own goal and Germany then conceded a goal when a ball saved by keeper Manuel Neuer bounced off Azerbaijan's striker Vagif Dzhavadov's back into the goal.
"We're having fun," said Philipp Lahm, the captain while Michael Ballack still lacks match practice after a three-month injury break.
"We had a short phase in the second half where we made way too many errors but all in all everyone can be happy."
