Croatia emerged from their third match at Russian Federation 2018 with their 100% record intact, as two second half goals canceled out an Icelandic penalty sends them into the knockout stages as Group D winners.
Sigurdsson also forced Lovre Kalinic into a save from a free-kick, but it was Cardiff midfielder Aron Gunnarsson whose curling drive from the edge of the area had him at full-stretch just before the break.
Perisic, with Luka Modric the only player not rested, struck in the last minute to set up a round-of-16 game against Denmark after Milan Badelj's second-half goal was cancelled out by Gylfi Sigurdsson's penalty.
Croatia broke the deadlock in the 53rd the minute and if the opportunity was something of a fluke, the finish most certainly wasn't.
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But Iceland got one back on a penalty kick in the 76th minute, as Croatia conceded for the first time in Russian Federation 2018.
But Iceland responded with intensity and intent, with a Sverrir Ingason needing touching over, and moments later he too smacked the crossbar.
Sigurdsson had ballooned a penalty over the bar against Nigeria, but this time dispatched his shot into the roof of the net with Kalinic diving the wrong way.
Iceland and Croatia know each other well, with coach Heimir Hallgrimsson likening the pair to old flames due to the number of times they have met in recent years. Perisic flew into the left of the box in the last minute and powered a shot through the hands of Hannes Halldorsson to seal their fate.
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Ivan Perisic has scored three World Cup goals for Croatia - only Davor Suker (6) has scored more. Iceland's maiden World Cup campaign, meanwhile, has ended with a first-round departure.
Ten minutes into the second period the match turned when Fiorentina striker Milan Badelj first clipped the bar with a long-range drive, then made no mistake a minute later.
He did the dirty work too, won half of his duels (7/14) and both of his tackles as well as recovering the ball more than anyone else on the pitch.
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