Arnautovic gamble pays off for Austria
Arnautovic's Werder Bremen team-mate Sebastian Proedl scored the other to leave Austria with six points from their opening, and arguably easiest, two games in Group A.
Former Germany and Scotland coach Berti Vogts' Azerbaijan have lost both their games so far.
The Austrians had struggled to a 2-0 home win against Kazakhstan last month, scoring boths goals in stoppage time.
This time the did it the easy way with Proedl heading them in front from a corner in the third minute.
Stefan Maierhofer was twice denied by Kamran Agayev before halftime as the Austrians dominated the game.
Arnautovic, whose allegedly lax attitude was criticised during an unhappy year with Inter Milan last season, opened his account eight minutes after the restart, neatly controlling Maierhofer's pass before scoring with a low shot from 15 metres.
The lanky 21-year-old added another in stoppage time with a cheeky chip over Agayev.