Michael Ballack opened the scoring with a penalty in the 14th minute, but Germany coach Joachim Low would not have been happy with the remainder of the first half as his side risked conceding an equaliser.
Their guests had Samir Abbasov sent off early in the second half and there was no stopping Germany with a one-man advantage. Two goals from Miroslav Klose and a Lukas Podolski strike sealed the comprehensive win which keeps Germany top of qualifying Group Four by a single point from Russia, who they meet in Moscow on 10 October.
After making his full debut in the 2-0 win over South Africa at the weekend, Mesut Ozil's name has become one of the first on Low's teamsheet and he showed precisely why he is one of Germany's hottest properties at the moment.
He left three Azerbaijan defenders stranded as he moved into a position to shoot in the seventh minute, but Kamran Agayev made his first save of a very busy opening 15 minutes.
The visiting goalkeeper was called into action five minutes later by Ballack, who tried his luck from long range before he was given the chance to open the scoring from the penalty spot.
Podolski was fouled inside the area by Samir Abbasov and the Chelsea midfielder stepped up to send Agayev the wrong way from the spot and give Germany a 14th-minute lead.
The floodgates were expected to open and Mario Gomez should have added a second two minutes later when he had a free header from Ballack's cross, but Agayev made another save.
However, Germany lost their early momentum too quickly and even risked conceding a goal before the interval when Olexandr Chertogan tested Rene Adler from long range. Adler tipped his shot over the crossbar before making a more difficult save to deny Mahir Shukurov moments later.
Low brought Klose on at the interval and he made an instant impact, luring Abbasov into a foul which earned the Azerbaijan captain his second yellow card of the evening in the 50th minute before the Bayern Munich striker doubled Germany's lead five minutes later.
He picked up the ball from Philipp Lahm before picking his spot past Agayev for a goal which will come as a major fillip to a player who no longer has a guaranteed place for club or country. The goal certainly did do Klose good, as he showed in the 66th minute when he provided an excellent finish to a deflected cross from Andreas Beck from just inside the six-yard box.
Podolski got in on the action five minutes later when he added the fourth after Ballack nodded Ozil's cross down to him inside the penalty area and Germany's first-half struggles had been forgotten as they looked capable of scoring on every attack.
Klose was denied his hat-trick when another effort was correctly ruled out for offside in the 73rd minute, but his two earlier goals had done enough damage to earn Germany a vital three points.