The Inter Milan attacker scored early in both halves, either side of a first-half Alexander Sorloth equaliser, before Philipp Lienhart, Stefan Posch and Michael Gregoritsch completed a dominant win for Austria.
Arnautovic put the hosts in front on eight minutes with a hammered strike from 20 yards that flew in off the crossbar.
Sorloth nipped in to head a leveller for Norway six minutes prior to the interval.
But the Austrian skipper put his side back in front in the 49th minute when he dispatched a penalty-kick after Christoph Baumgartner was fouled in the box.
Defenders Lienhart and Posch extended the lead with two towering headers two minutes either side of the hour mark.
Gregoritsch then scored a third header of the half for the Austrians as they made the game safe and moved second in Group B3.
Manchester City star Haaland had a night to forget with the Austrian defence comfortably subduing the Norwegian captain whose only real chance was a shot which came back off the post in the fifth minute.
Slovenia sit third in the group, although level on seven points with both Norway and Austria, after winning 1-0 in Kazakhstan courtesy of a second-half goal from forward Jan Mlakar.
On Monday, Belgium host 2021 Nations League champions France while old rivals Germany and Netherlands clash in Munich.
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