Only Perisic and Luka Modric kept their places in Croatia's starting XI for their final group match against Iceland on Tuesday evening.
The Balkan side could afford to make the changes having already secured qualification while their opponents were still fighting to secure theirs.
Despite Iceland needing to secure a victory, they were rather lathergic in the first half and didn't spring into action until the dying moments.
However, Alfred Finnbogason's shot whistled past the post to ensure that both sides went in goalless at the break.
Iceland had to go out and attack in the second and this allowed Croatia to find their breakthrough in the 53rd minute.
A series of fortunate deflections meant the ball fell in the path of Milan Badelj, who controlled his volley beautifully before sending it crashing into the back of the net from 10 yards.
It left the island nation stunned and they only found their way back into the match when they were handed a lifeline 20 minutes later.
Substitute Dejan Lovren moved his arm toward a Sigurdsson shot, leaving the referee no choice but to point to the spot.
Despite missing against Nigeria, Sigurdsson stepped up to take the spot-kick and made no mistake with his effort, going high in the middle to bring Iceland level.
It meant they only needed one more goal to secure progression to the Round of 16 but a series of substitutions handed momentum back to Croatia.
They took full advantage in the final minutes and secured a late victory thanks to a strike from Perisic, which opposing goalkeeper Harve Halldorsson could only parry into the top corner.
It will be a difficult defeat for Iceland to swallow given the stakes, but they will have won numerous admirers in Russia for their dogged determination.
Unfortunately, this summer was a step too far but, for Croatia, the future looks very bright. They finish Group D in first position with a perfect record and now face Denmark in the last 16.