The Italians went ahead when Sampdoria striker Eder slotted in from Marco Verratti's superb pass.
Azerbaijan equalised through Dmitri Nazarov's low, first-time strike, but Italy regained the lead as Stephan El Shaarawy tapped in from close range.
Manchester United full-back Matteo Darmian hit a third, and Azeri defender Badavi Huseynov was sent off late on.
Having failed to make it through to Sweden in 1992, Italy - runners-up in 2000 and 2012 - have now qualified for six consecutive Euro tournaments.
Norway beat Malta 2-0 to go five points clear in second place, with third-placed Croatia facing Bulgaria later on Saturday.
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