Milan Badelj hit a sweet half-volley for his side early in the second half, and Ivan Perisic struck in a 90th-minute victor for Croatia after Gylfi Sigurdsson had pulled Iceland level from the penalty spot.
Badelj didn't have to wait long though as two minutes later he found himself unmarked in the box to hammer in a cross from Pivaric to give Croatia a 1-0 lead.
A win or a draw against Iceland clinches first place for Croatia, and they still likely would top the group even if they lost and Nigeria won because the African nation would need to make up a five-goal deficit in the goal-differential tiebreaker.
Argentina took second place and will face France after a 2-1 win against Nigeria. Soccer: World Cup, Argentina vs Croatia, preliminary round, Group D, 2nd match day in the Nizhny Novgorod Stadium.
It started to press higher, putting Croatia on the backfoot.
For all of Croatia's dominance, Iceland did pose a real threat on the counter attack but they were let down by their decision-making in the final third of the pitch. Croatia, with most players looking out of match form, were lucky to go into the break level at 0-0. Sigurdsson made no mistake from the spot. They were then in the same group for 2018 qualifying, where they split the head-to-head results, but Iceland finished on top to seal the automatic berth.
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Both sides pushed for a victor with Dalic introducing Rakitic for Kovacic.
A revamped but businesslike Croatia topped their World Cup group as Ivan Perisic's 90th-minute goal helped them maintain their flawless record with a 2-1 win on Tuesday that also ended Iceland's qualification hopes.
Alfred Finnbogason missed wide in the 40th minute after his own takeaway, Emil Hallfredsson skied it over the crossbar from 33 yards four minutes later and seconds from the half-time whistle, Aron Gunnarsson's attempt from 20 yards was batted away by goalkeeper Lovre Kalinic.
"It was a hard match and very humid". It was a deserved win in the end but it was not easy.
GOALS: Badelj 53′, Sigurdsson 71′ (P) Perisic 90′.