The West Ham forward showed plenty of composure to break the deadlock in the 70th minute on Friday night, scoring past Bailey Peacock-Farrell in the Northern Ireland goal after finding space down the right behind former Manchester United player Paddy McNair.
In a game short on real quality, Arnautovic's perfect first touch and calm finish was enough to see the Alpine nation, ranked 24th in the world by FIFA, cement their spot in second place behind Bosnia-Herzegovina in Group B3, reports Efe news.
Northern Ireland prop up the group, having failed to win either of their games so far.
The visitors were limited to opportunities from set pieces and throw-ins, although they should have scored when captain Steven Davies was clean through, but he failed to capitalize on Sebastian Prodl's mistake.
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