The visiting Scandinavians dominated the opening exchanges at the Astana Arena, but it wasn't until the 20th minute that they were able to break the deadlock.
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It was the former Chelsea and Barcelona frontman's first international goal since 2009 and at the age of 36, he also became the fourth-oldest goalscorer in Euro qualifiers behind Jari Litmanen, John Aldridge and Krasimir Balakov.
During what was a largely uneventful second half, Kazakhstan defender Rinat Abdulin thought that he had reduced the arrears in the 63rd minute, but his headed effort struck the post before the ball was hacked to safety.
It proved to be the only nervy moment of note for Lars Lagerback's Iceland, who added a third goal in stoppage time courtesy of Bjarnason's second strike of the contest.
The victory means that Iceland sit six points clear of the Netherlands in second spot in Group A, having won four of their five qualification encounters.