Norway will be disappointed to drop points at the start of a qualifying campaign in which they have high hopes of reaching the finals in Brazil in two years.
The visitors started well and would have taken the lead after 17 minutes but for a superb save from Iceland goalkeeper Hannes Thor Halldorsson to deny Mohamed Abdellaoui.
But Iceland were ahead just four minutes later as Rotherham midfielder Arnason finished from close range following Aron Gunnarsson's cross.
Norway had a number of set-pieces that did not lead to any clear-cut opportunities and it took until the hour mark to really test Halldorsson again.
Bjorn Helge Riise forced a smart save out of the KR Reykjavik goalkeeper before Iceland came close to a second goal.
Helgi Danielsson collected the ball following a Gretar Steinsson cross and brought a very good save out of Norway goalkeeper Espen Pettersen.
But there was nothing Pettersen could do to prevent Finnbogason doubling the host`s lead with a little under ten minutes remaining as the 23-year-old was picked out by Tottenham's Gylfi Sigurdsson and finished cooly from inside the box.
Manchester United striker Joshua King came close to pulling a goal back for Norway in injury time but his long-range effort came back off the post.