Against a side whose game plan was simply to defend at all costs, it still took only 11 minutes for Croatia to forge ahead as Andrej Kramaric converted from close range after a precise cross from Darijo Srna.
Perisic, on the substitutes' bench, may have been expecting to miss out on any goal deluge but was given an unexpected early chance to shine when Mateo Kovacic had to come off injured after 24 minutes.
Ten minutes later, Vfl Wolfsburg winger Perisic headed home Ivan Rakitic's free kick and, in the dying moments of the first half as Azerbaijan's defence completely fell apart, he made it three with a fine header from Danijel Pranjic's cross.
Two minutes into first half stoppage time, Luka Modric's brilliant pass set up Marcelo Brozovic to make it four before Modric himself converted a penalty after the break following a foul on Srna.
An own goal by Rashad Sadygov completed Azerbaijan's misery just after the hour mark, leaving Croatia to contemplate a much more meaningful encounter when they meet Italy, also unbeaten on nine points, in Milan next month.
(Writing by Igor Ilic in Zagreb; Editing by Ian Chadband)
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