Gylfi Sigurdsson's 76th-minute penalty gave Iceland hope of mounting a comeback following Milan Badelj's opener, but Perisic made sure of the points for Croatia in the dying embers.
Croatia will now meet Denmark in the round of 16 in Nizhny Novgorod on Sunday, while Iceland's maiden appearance at a World Cup comes to an end despite their best efforts to mount a comeback.
Badelj - who had previously rattled the woodwork - marked his first start of the tournament with a thumping 53rd-minute opener.
Sverrir Ingason rattled the bar with a header in response for Iceland, but the only way they could beat Croatia goalkeeper on his superb World Cup bow came from the spot.
Sigurdsson missed a penalty in Iceland's defeat to Nigeria on matchday two, but made no mistake this time with a finish high into the net after Dejan Lovren had handled in the area.
But Iceland's fighting spirit ultimately proved for nothing, Perisic securing a third successive win for Croatia, who will be joined in the last 16 by Argentina following their dramatic win over Nigeria.
