Croatia clinched its place in the last 16 with a 2-0 win over Nigeria and a statement 3-0 win over Argentina.
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.
Croatia had their first clear chance when Milan Badelj's missile crashed onto the bar six minutes into the second half.
Croatia is in a great position to clinch the top spot in Group D at the 2018 FIFA World Cup in Russian Federation.
Croatia continued to threaten, however, with Iceland starting to appear worn out, and Perisic finished off the win in the 90th minute, scoring left-footed from 12 yards to secure the victory. Johann Berg Gudmundsson stole possession before trying to set Alfred Finnbogason away, only to be denied by an interception from Dulje Caleta-Car.
They started to press higher, putting Croatia on the back foot.
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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.
Iceland will have to copy their game plan that worked in their win over Croatia.
Hallgrimsson cut his teeth in worldwide management as assistant to Swede Lars Lagerback, who steered Iceland to the Euro 2016 finals, where they beat England in the last 16 but fell to host nation France at the quarter-final stage.
Iceland nearly responded instantly to the setback through centre-back Ingason, whose two quick-fire headers first drew a smart stop from Kalinic, before being denied by the crossbar moments later having met a corner unmarked.
Iceland and Croatia were in the same qualifying group for the World Cup and landed in the same group at the World Cup.
The debutants finally grabbed the goal that their industry perhaps deserved with just under 15 minutes to play, as substitute defender Dejan Lovren was penalised for a handball that blocked Sigurdsson's cross. The midfielder was more accurate two minutes later as he thundered a volley into the back of the net to open the scoring.