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Martin Harnik put the group leaders ahead on 14 minutes when he collected a clever back-heel on the edge of the Liechtenstein box and drove into the area to fire home.
The lead should have been three just after the half-hour mark when Liechtenstein goalkeeper Peter Jehle conceded a penalty with a rash challenge.
However, Bayern Munich's normally reliable David Alaba stepped up and stroked the penalty wide of Jehle's right-hand post.
Alaba made amends on 60 minutes when he collected the ball 25 yards from goal, turned away from danger and sent a curling strike beyond the outstretched dive of Jehle.
Zlatko Junuzovic made it four and Arnautovic completed the rout late on to extend Austria's unbeaten run that keeps them firmly on course for Euro 2016 qualification.