The Italians, who were runners-up to champions Spain in the 2012 finals, secured their place in next year's finals in France by maintaining their unbeaten run in the group with six wins and three draws from their nine matches.
They have 21 points, followed by Norway on 19 and Croatia on 14, with the Croatians playing Bulgaria later on Saturday.
Eder opened the scoring for the visitors after 11 minutes when he collected an incisive through ball from Marco Verratti behind the Azerbaijan defence and slid the ball home past Kamran Agayev.
Already-eliminated Azerbaijan equalised 20 minutes later when mistakes by Giorgio Chiellini and Leonardo Bonucci allowed Dmitri Nazarov to score.
Italy regained the lead two minutes before the break when a good move involving Graziano Pelle and Antonio Candreva ended with Stephan El Shaarawy planting the ball into an empty met.
The third Italian goal came in the 65th minute when Matteo Darmian took advantage of a loose ball on the edge of the Azerbaijan box and beat Dmitri Nazarov to it before smashing home past Agayev.
Azerbaijan finished the match with 10 men after Badavi Hüseynov was sent off for pulling back Sebastian Giovinco denying him a scoring chance.
(Writing by Mike Collett; Editing by Rex Gowar)
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