The minnows, playing in their first ever World Cup final, went behind with 18 minutes gone as Mario Mandzukic put through his own net, but despite France's quality, they were unable to hold onto their lead for long.
Just 11 minutes after the opener, Croatia levelled, with the goal also coming from a set-piece just as France's did.
Perisic was tripped by N'Golo Kante as he looked to burst through the middle and Luka Modric picked out Sime Vrsaljko's late run with the dead-ball.
The full-back headed the ball back into the mixer and it eventually made its way to Perisic on the edge of the box. He jinked a yard of space away from Kante before thundering a low effort beyond Hugo Lloris and into the bottom corner to restore parity.