In what was a one-sided encounter from start to finish, Croatia ran in four goals before the break en route to moving top of Group H after making it nine points from three games.
With 11 minutes played, the Croats took the lead when Andrej Kramaric directed Darijo Srna’s deflected cross high into the net from close range.
Ten minutes later there was almost a second goal as Danijel Pranjic smashed a volley against the woodwork, with keeper Kamran Agayev well beaten.
The visitors somehow managed to prevent Croatia from making it 2-0 until the 34th minute when Perisic superbly glanced home Ivan Rakitic free-kick.
On 44, Perisic headed the third from Pranjic’s cross and there was still time for another goal before half time.
Two minutes into injury time, Marcelo Brozovic fired home from 12 yards after latching on to Luka Modric’s pass.
In truth, Croatia could have been six or seven goals ahead by half-time such was their domination.
The second half was just as straightforward for the clinical Croats, with Modric firing home a fifth from the penalty spot in the 56th minute after Elnur Allahverdiyev’s high boot had caught Ivica Olic in the head.
And to make matters worse on the hour, skipper Rashad Farhad Sadygov put through his own net from seven yards after miscuing Perisic’s low cross.
Croatia nearly made it seven in the closing few seconds as 18-year-old substitute Alen Halilovic saw his effort bounce off teammate Olic and strike a post.
Azerbaijan, who will have been glad to hear the full-time whistle, are now bottom of the table with zero points from three games.