Just as he did when the two sides met in Vienna in 2014 World Cup qualifying, Alaba put the home side in front when he confidently fired home in the seventh minute after Sebastian Larsson handled.
Five minutes later Zengin struck a spectacular first international goal for Sweden, thrashing in a right-foot effort after being perfectly set up by Zlatan Ibrahimovic.
Having broken Sweden's all-time scoring record with a double against Estonia last week, captain Ibrahimovic was largely subdued on his 100th appearance, failing to unduly trouble the Austrian rearguard and wasting several good opportunities from free kicks.
Austria won a flurry of corners early in the second half but could not capitalise and the Swedes grew in confidence as they held on for the draw.
The stalemate means the Swedes have still never won in Austria but having been beaten 2-1 here on their last outing, coach Erik Hamren will be happy to have gained a point from one of the tougher away games his side face in qualifying.
(Reporting by Philip O'Connor, editing by Tony Jimenez)
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